Alvin R. Moore, Jr. was convicted in a Louisiana court of the murder of Jo Ann Wilson and sentenced to die. After exhausting his state remedies, Moore filed an application for federal habeas relief. The *311district court granted Moore partial habeas relief and ordered that he have a new penalty trial. The State of Louisiana now appeals the district court’s grant of partial habeas relief. Moore cross-appeals in order to preserve those issues raised in his habeas petition that were not addressed by the district court. For the reasons set forth below, we reverse that part of the district court’s judgment granting Moore relief.
I. FACTUAL AND PROCEDURAL BACKGROUND.
Aaron Wilson, his wife Jo Ann, and their four-month-old daughter Regina lived at 804 St. Charles Street in Bossier City, Louisiana.1 Alvin Moore was a former neighbor and co-worker of Aaron’s at the Veteran’s Administration Hospital. On July 9, 1980, Moore picked up Arthur Lee Stewart, Jr. and Dennis Sloan in. his automobile and the three rode around Shreveport, visiting, laughing and talking. Moore then said he wanted to go to Bossier City to “check on his old house.” Moore’s former residence was at 833 St. Charles Street.
After noticing that Aaron Wilson’s car was gone, Moore went to the Wilsons’ house with the announced purpose of getting “some money.” According to Sloan, Moore knocked on the door and Jo Ann Wilson answered it. She and Moore talked briefly and Moore entered the house. Five minutes later, Sloan followed Moore to the door of the house. The door was slightly ajar and Sloan saw Moore and Jo Ann Wilson “making sex” on the floor of the living room. Sloan returned to Moore’s automobile to tell Stewart what was happening. Stewart and Sloan then entered the house. Moore and a crying Mrs. Wilson had gone into the bedroom where baby Regina was also crying. Moore was “going crazy,” ransacking the house. Jo Ann Wilson was described as “panicky” and “scared.” She also appeared to be frightened of Moore. Sloan testified that Jo Ann Wilson said, “Take whatever you want, just get out of my house.” Sloan also testified that Mrs. Wilson asked Moore not to harm her or her child.
After being threatened, Mrs. Wilson gave Moore a box of Kennedy half dollars. Sloan took a white bucket with $18.80 in pennies, and Stewart took some stereo components. Sloan and Stewart left the house and heard Jo Ann Wilson screaming behind them.
Moore ran out of the house five minutes later carrying a knife in his hand. Stewart testified that this was the same knife that Moore had had on the back seat of his car when the group drove to St. Charles Street. Moore told Stewart and Sloan, “Pm fixing to trip you all out ... I stabbed that bitch nine times.” The three then drove to Church’s Fried Chicken and McDonald’s.
Jo Ann Wilson managed to call the police emergency number. The call was received by the Bossier City Police Department at 9:40 p.m. and a unit was dispatched. Patrolman Fields arrived at the house two minutes later. He knocked on the front door, but Jo Ann Wilson said she was unable to open it. He kicked in the door and found blood all over the living room. Officer Fields found Mrs. Wilson lying on the bed in the bedroom. Both rooms were in disarray. The victim was nude from the waist down, and was bleeding from her vagina, chest, and arms. She was having difficulty breathing, and told Officer Fields that she was dying. He asked her who stabbed her and she responded, “Elvin.” Fields asked her if she knew “Elvin,” and she replied that he “was a black guy that used to live down the street.” It was obvious to Fields that Mrs. Wilson was dying, and she died approximately ten minutes after her dying declaration. Moore was arrested at 1:00 a.m. the next morning. The Wilsons’ stereo components and the white bucket and pennies were found in the trunk of Moore’s car.
Dr. George McCormick, coroner of Bossier Parish and a forensic pathologist, performed an autopsy on the victim. At the guilt phase of the trial, he testified that Jo *312Ann Wilson had received thirteen stab wounds and three other minor wounds. She was wounded on her shoulders, arms, chest, back, and in the pelvic and abdominal areas. There was also testimonial evidence that Mrs. Wilson had recent vaginal and rectal intercourse, probably within six to eight hours of the autopsy.
Blood flakes taken from Moore’s pants were identified as blood group 0. Mrs. Wilson had blood group 0; Sloan, Stewart and Moore had blood group B. No blood was found on the clothing of either Stewart or Sloan. Foreign pubic hairs taken from Jo Ann Wilson’s vagina did not match those of Sloan, Stewart or her husband, but matched the pubic hair characteristics of Moore. One hair on the bed sheet also matched all the characteristics of Alvin Moore’s pubic hair.
Moore took the stand in his defense. He contended that his intercourse with Jo Ann Wilson was voluntary, and that he had had voluntary intercourse with her on two previous occasions in May and June. In May, Mrs. Wilson had also given him some money. Moore testified that he did not take the Kennedy half dollars from Mrs. Wilson; rather, she had given them to him. Moore denied any knowledge of the stabbing.
Moore was found guilty of first degree murder. At the sentencing phase both sides reintroduced all of the evidence that had been submitted at the guilt phase. The jury found three aggravating circumstances: (1) That Moore was engaged in the commission or attempted commission of aggravated rape, aggravated burglary and armed robbery; (2) that Moore knowingly created a risk of death or great bodily harm to more than one person; and (3) that the offense was committed in an especially heinous, atrocious or cruel manner.2 The jury recommended the death penalty.
On appeal, the Louisiana Supreme Court upheld Moore’s conviction and sentence. State v. Moore, 414 So.2d 340 (La.1982). The court found, however, that there was insufficient evidence to support the jury’s finding that Moore had created a risk of death or great bodily harm to baby Regina.3
Moore petitioned for a writ of certiorari to the United States Supreme Court, and the Court stayed Moore’s execution pending disposition of his petition. Moore v. Louisiana, — U.S. ---, 104 S.Ct. 38, 77 L.Ed.2d 1456 (1983). On June 27, 1983, the Supreme Court denied Moore’s petition for a writ of certiorari, and Moore was scheduled to be executed on August 11, 1983. Moore petitioned for habeas relief in state district court, which was denied. Moore then sought habeas relief in the Louisiana Supreme Court, which was also denied. Moore v. Maggio, 435 So.2d 997 (La.1983).
Having exhausted his state remedies, Moore then filed a petition for federal habeas relief and a motion for a stay of execution in the court below on August 9, 1983. The district court issued an order staying Moore’s execution until the court could pass on Moore’s habeas petition.
In his habeas petition, Moore contended that he was entitled to relief because:
(1) He received ineffective assistance of counsel at the guilt phase of the trial;
(2) He received ineffective assistance of counsel at the penalty phase of the trial;
(3) The Louisiana Supreme Court failed to engage in a meaningful appellate review designed to ensure that death was the appropriate sentence;
*313(4) One of the three aggravating circumstances found by the jury was unsupported by the evidence;
(5) The trial court’s exclusion of jurors who were unambiguously opposed to imposing the death penalty resulted in a biased and unfair jury;
(6) The trial court gave the jury inadequate instructions concerning imposition of the death penalty; and
(7) The death penalty in Louisiana, as well as in the United States, is applied in a racially discriminatory and arbitrary manner.
On October 17-18, 1983, the district court held an evidentiary hearing concerning Moore’s claim of ineffective assistance of counsel at the penalty phase of his trial. Following the hearing, the court concluded that Moore had been denied effective assistance of counsel during that part of the trial. The court vacated Moore’s death sentence and ordered a new sentencing trial. The district court found that Moore received effective assistance of counsel at the guilt phase of the trial, a determination that Moore does not appeal. The court found it unnecessary to address Moore’s remaining contentions because they related to Moore’s sentence, which was vacated by the court.
The State of Louisiana now appeals the district court’s order granting Moore a new sentencing trial. In the event that we reverse the district court’s grant of partial habeas relief, Moore contends that the case should be remanded so that the district court can consider issues (3) through (7), which the court declined to address in the prior proceeding.4
il. INEFFECTIVE ASSISTANCE OF COUNSEL AT THE PENALTY HEARING.
In the court below, Moore challenged counsel’s assistance during the penalty phase in four respects. He asserted that counsel was ineffective because (1) he failed to present evidence of mitigating circumstances to persuade the jury not to recommend the death penalty; (2) he failed to argue or plead for Moore’s life; (3) he did not during the voir dire attempt to identify and select jurors who were inclined toward life imprisonment rather than death; and (4) he failed to object to the trial court’s allegedly inadequate jury instructions regarding the imposition of the death penalty.5
A. The District Court’s Findings and Conclusions.
As we noted earlier, the district court held an evidentiary hearing with regard to Moore’s claim that he received ineffective assistance of counsel at the penalty hearing. At the hearing, both Moore’s counsel and the attorney who assisted him testified, as did Moore’s parents, minister, and girlfriend. Moore produced several expert witnesses who testified as to the inadequacy of Moore’s counsel’s performance. Moore also introduced the affidavits of twenty-three individuals who attested as to Moore’s good character.
Following the hearing, the district court issued a carefully crafted opinion that contained several factual findings: First, the court found that although counsel interviewed both Moore and his parents about *314Moore’s childhood and religious activities, counsel interviewed no other potential mitigation witnesses. These other potential mitigation witnesses included Moore’s pastor and Moore’s girlfriend, who was carrying Moore’s child at the time.
The court found that counsel decided not to introduce mitigation witnesses who would testify as to Moore’s good character because of incriminating evidence that counsel discovered that the prosecution could use to impeach Moore’s character. For instance, counsel learned that while in school, Moore struck a janitor in the head with a pipe wrench. Counsel also discovered that Moore had been convicted of assaulting a store owner, and that Moore had shot out the windows of a car with a shotgun. Counsel also learned that Moore had beaten and robbed a sixty-eight year old woman in a Veteran’s Administration Hospital parking lot.
Although counsel could not remember how he came to learn about each one of these incidents, he testified that members of the district attorney’s office informed him of the mugging in the hospital parking lot, and that counsel had seen the police report on the incident.
At the guilt phase of the trial, counsel’s strategy was to convince the jury that there was a reasonable doubt as to Moore’s guilt. Counsel attempted to shift the blame to Sloan and Stewart, Moore’s co-defendants, before they pleaded guilty to reduced charges. This strategy was prompted, in part, by Moore’s repeated protestations of innocence, and by Moore’s insistence on taking the stand. At the close of the guilt phase, counsel gave a lengthy argument to the jury describing in great detail the reasons why he believed that Moore was innocent. The jury returned its guilty verdict and, after a short recess, the penalty phase of the trial began. After the prosecutor argued why he thought the death penalty was appropriate in this instance, Moore’s counsel addressed the jury:
Ladies and gentlemen, it's, of course, a little difficult for me to stand here feeling the way I do about this case, and now ask you to consider mitigating circumstances in the sentence. There are some mitigating circumstances that you can consider, there are aggravating circumstances, you’ll be given lists of both of those by the Court that you’ll consider. You’ve got to consider at this point whether to take a human life for what has happened, that you feel has been proved. You’ve got to consider some mitigating circumstances such as the age of the offender, other things that will be given you. Just based on these things, you’ve got to — you’ve got to come to some sort of conclusion. As I say I find it hard, I don’t know at this point exactly how — what to say about it when I feel as strong as I do about the situation, to be standing here saying what Fm saying. I hope that you do consider these circumstances, hope you consider them very seriously and think about for a moment what you have the potentiality of doing. I hope you think about it now rather than a couple of weeks from now or tomorrow or the next day, because we’ll be thinking about this case for a long time after this. And I hope that — but your time to think and make a rational judgment is now. The mistake can be corrected, no doubt, but I ask you to give it your most serious consideration in the next short length of time, and determine that the death penalty would not be proper in this case. Thank you.
Joint Exhibit 1C at 776-77.
From counsel’s testimony at the evidentiary hearing, the district court found that, following the guilt phase, counsel remained convinced of Moore’s innocence and that counsel’s argument to the jury was that it should not vote for the death penalty in the face of doubts he had outlined in his closing argument at the guilt phase. However, based on its own reading of the transcript of counsel’s closing argument, and the testimony of Moore’s expert witnesses, the court found that counsel failed to plead for Moore’s life or to ask the jury not to impose the death penalty.
*315The district court concluded that counsel had no strategy for the penalty phase of Moore’s trial, that counsel failed to conduct a meaningful search for mitigation witnesses, and that counsel did nothing to convince the jury that it should not recommend the death penalty. The district court held that counsel’s deficiencies deprived Moore of effective counsel and that these deficiencies were sufficiently prejudicial that Moore was entitled to a new sentencing trial.
B. Strickland v. Washington.
Subsequent to the district court’s ruling, the Supreme Court decided Strickland v. Washington, — U.S. ---, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984), which enunciates the standards to be applied when reviewing a claim of ineffective assistance of counsel. There, the Court held:
A convicted defendant’s claim that counsel’s assistance was so defective as to require reversal of a conviction or death sentence has two components. First, the defendant must show that counsel’s performance was deficient. This requires showing that counsel was not functioning as the “counsel” guaranteed the defendant by the Sixth Amendment. Second, the defendant must show that the deficient performance prejudiced the defense. This requires showing that counsel’s errors were so serious as to deprive the defendant of a fair trial, a trial whose result is reliable. Unless a defendant makes both showings, it cannot be said that the conviction or death sentence resulted from a breakdown in the adversary process that renders the result unreliable.
104 S.Ct. at 2064.
Before we determine whether Moore made such a showing in this case, we believe it would be helpful to delineate the practical application of Strickland v. Washington’s two-pronged test. With regard to the deficiency prong, the Supreme Court held that “[¡judicial scrutiny of counsel’s performance must be highly deferential.” 104 S.Ct. at 2065. Our assessment of counsel’s performance requires us to make every effort to eliminate the distorting effects of hindsight and to evaluate the conduct from counsel’s perspective at the time. Because of the difficulties of assessing counsel’s performance, the Court has directed us to “indulge a strong presumption that counsel’s conduct falls within the wide range of reasonable professional assistance; that is, the defendant must overcome the presumption that, under the circumstances, the challenged action ‘might be considered sound trial strategy.’ ” 104 S.Ct. at 2066 (citation omitted).
An error by counsel, even if professionally unreasonable, does not require setting aside a defendant’s conviction or sentence if the error had no effect on the judgment. Thus, it is insufficient for the defendant to show that counsel’s errors had some possible effect on the verdict. “The defendant must show that there is a reasonable probability that, but for the counsel’s unprofessional errors, the result would have been different.” 104 S.Ct. at 2068. The Court held that when a defendant challenges a death sentence, “the question is whether there is a reasonable probability that, absent the errors, the sentencer — including an appellate court, to the extent it independently reweighs the evidence — would have concluded that the balance of aggravating and mitigating circumstances did not warrant death.” 104 S.Ct. at 2069. We now turn to apply these principles in the instant case.
C. Deficiency Prong.
The district court held that Moore’s counsel did not adequately prepare for the sentencing phase because he did not seek any mitigation witnesses other than Moore’s parents and Moore himself. In the circumstances of this case, we do not believe that counsel failed to conduct a reasonable investigation of Moore’s background. Moore’s mother testified that she informed counsel that her son had been expelled from school and that he had been involved in fights. She also told counsel that Moore had been convicted of shoplifting and that he had an erratic employment *316record. Counsel also learned from Moore and his parents that Moore had stopped going to church for more than two years. Counsel could reasonably surmise from his conversations with Moore and his parents that, in view of the aggravating circumstances, character evidence would be of little help.6
Although counsel did not interview Moore’s girlfriend, he decided that it would not be wise to call her as a witness because she was carrying Moore’s child at the time. Counsel also decided not to put Moore’s parents on the stand because the prosecutor would have then introduced evidence of Moore’s prior criminal activities. Counsel testified that he discussed this damaging evidence with the prosecutor and that counsel agreed not to put on character evidence on Moore’s behalf if the state would not introduce the evidence of Moore’s prior criminal and antisocial behavior. In counsel’s estimation, the harm that would result from the prosecutor’s rebuttal evidence outweighed the potential value of favorable character evidence. Moreover, Moore’s parents attended trial every day, and had been openly affectionate and supportive of their son in front of the jury. Counsel believed that this would help to convince the jury that Moore had the love and support of his family, without implicating the risks that were inherent in having the parents testify. In these circumstances, we believe that counsel’s strategy fell within the range of professionally reasonable judgment.
The district court found that counsel made no plea for Moore’s life at the penalty hearing. This factual determination was based on the court’s reading of counsel’s closing argument, quoted supra. We have often held that where the district court’s finding is based on documentary evidence, the “clearly erroneous” standard still applies. However, where factual findings are based on documentary evidence and where credibility determinations are not seriously involved, the presumption of correctness accorded the district court’s findings under the clearly erroneous rule is lessened. Burston v. Caldwell, 506 F.2d 24, 26-27 (5th Cir.), cert. denied, 421 U.S. 990, 95 S.Ct. 1995, 44 L.Ed.2d 480 (1975); see also Onaway Transportation Co. v. Offshore Tugs, Inc., 695 F.2d 197, 200 (5th Cir.1983); Emmco Insurance Co. v. Wallenius Caribbean Line, S.A., 492 F.2d 508, 512 (5th Cir.1974).
At the hearing, Moore’s expert witnesses testified that, based on their readings of the transcript, they did not bélieve that counsel had asked the jury to spare Moore’s life. We believe, however, that an equally plausible reading of the transcript could be that counsel did indeed reiterate his belief that Moore was innocent, and that the jury would be making a mistake if it recommended the death penalty for Moore. If our assessment of the district court’s finding was based solely upon the court’s reading of the document itself, and the expert testimony construing it, we would hesitate to say that “we are ‘left with the definite and firm conviction that a mistake has been committed.’ ” Burston v. Caldwell, 506 F.2d at 27 (citation omitted). However, our review of the district court’s findings does not rest solely on the documentary evidence.
Whether counsel pleaded for Moore’s life at the penalty hearing must be assessed in light of the manner in which counsel delivered his closing argument. At the evidentiary hearing held in the court below, the attorney who assisted Moore’s counsel at trial testified that counsel was very emotional when he delivered his closing argument and that counsel “came close to *317tears.” Counsel’s assistant testified that counsel pleaded for Moore’s life and emphasized the finality of the jury’s decision. Counsel himself testified that, although his argument was short, he had just delivered a lengthy closing argument at the guilt phase, in which he pleaded with the jury not to convict Moore. Counsel had explained in great detail why he believed there was a reasonable doubt as to Moore’s culpability, and he did not want to “insult the jury” by reiterating those reasons little more than an hour later. Thus, it appears that, based on the testimony of those who were present at the trial, counsel argued to the jury that he still believed in Moore’s innocence and, in light of the asserted reasonable doubt as to Moore’s guilt, the jury should not impose the irrevocable penalty of death.
Counsel and his assistant were the only two witnesses who testified as to what transpired at the trial. Although Moore’s expert witnesses construed counsel’s closing argument as not asking the jury to spare Moore’s life, their interpretation was based solely upon a reading of the trial transcript, and not as a result of having witnessed the closing argument at the penalty hearing. In finding that counsel made no plea for Moore’s life, the district court’s determination rested upon the experts’ interpretation of the trial transcript, and the court appeared to pay little heed to uncontradicted testimony as to the manner and tone in which counsel’s argument was delivered. Because we believe that the text of counsel’s argument is susceptible of more than one meaning, we find that testimony as to how the argument was delivered assumes a critical role. Of course, the district court was not required to accept either counsel’s or his assistant’s testimony as to what occurred at trial, even though neither one of these witnesses was contradicted. Burston v. Caldwell, 506 F.2d at 26; Goodwin v. Smith, 439 F.2d 1180, 1182 (5th Cir.1971). Yet there is no indication that the district court found their testimony not credible and, based on this uneontradieted testimony as well as our reading of counsel’s closing argument at both the penalty and guilt phases of Moore’s trial, we find that the district court erred in concluding that counsel did not plead for Moore’s life.
The district court also found that counsel “did not argue against the death penalty on moral, ethical or religious grounds,” and thus concluded that counsel did nothing to convince the jury that it should not impose the death penalty. At the evidentiary hearing, however, counsel testified that he did not believe that an argument against the death penalty couched in moral or religious terms would have been effective in front of that particular jury, a judgment we will not second-guess.
Moore also contends that counsel failed to attempt to rehabilitate any of the four veniremen that the prosecutor successfully challenged because they unequivocally stated that they would not vote to impose the death penalty in any circumstances. See Witherspoon v. Illinois, 391 U.S. 510, 88 S.Ct. 1770, 20 L.Ed.2d 776 (1968). See O’Bryan v. Estelle, 714 F.2d 365, 377-78 (discussing rehabilitation of veniremen challenged on Witherspoon grounds), cert. denied, — U.S. -, 104 S.Ct. 1015, 79 L.Ed.2d 245 (1984). Our review of the record establishes that counsel objected to the prosecution’s challenge of these veniremen until he had had a chance to question them. Despite the unmistakable clarity with which each venireman expressed his or her opposition to the death penalty in any circumstances, counsel attempted with no success to rehabilitate three of the four. With reference to the fourth, we do not consider counsel’s decision not to attempt rehabilitation, in light of the venireman’s unequivocal opposition to the death penalty, to have constituted ineffective assistance of counsel.
Moore also complains that counsel failed to explore the jury’s racial attitudes in light of the fact that the victim was white and the defendant was black. Although the trial court must afford a defendant the opportunity to question venire*318men under certain circumstances, Ristaino v. Ross, 424 U.S. 589, 96 S.Ct. 1017, 47 L.Ed.2d 258 (1976); Ham v. South Carolina, 409 U.S. 524, 93 S.Ct. 848, 35 L.Ed.2d 46 (1973), whether the defense will avail itself of such an opportunity will in most instances involve the exercise of a judgment that should be left to counsel. Gustave v. United States, 627 F.2d 901, 906 (9th Cir.1980). In the instant case, the trial was not attended by any racial animosity, and we do not believe that Moore has overcome the presumption that, in the circumstances, counsel’s decision not to bring up the subject of racial bias before the jury “ ‘might be considered sound trial strategy.’ ”7 Strickland v. Washington, 104 S.Ct. at 2066 (quoting Michel v. New York, 350 U.S. 91, 101, 76 S.Ct. 158, 164, 100 L.Ed. 83 (1955)).8
D. Prejudice Prong.
Even if we were to assume that counsel’s performance at the sentencing phase was deficient, Moore has failed to show that he was prejudiced. At the penalty hearing, the prosecution reintroduced the evidence submitted at the guilt phase that detailed the horrifying nature of Moore’s crime. At the guilt phase, there was medical testimony that the victim had received multiple wounds all over her body, and other testimony established that the victim died slowly, with awareness of her impending death. There was also evidence from which the jury could infer that the victim was vaginally and anally raped.
Moore complains that, for example, counsel should have called his minister as a character witness; however, at the evidentiary hearing held in the court below, Moore’s minister not only testified that Moore had not been attending church, but also that he was unaware of Moore’s prior criminal record. The twenty-three affidavits that Moore introduced were from friends, neighbors, and relatives, and generally alleged that Moore was a “likeable person,” and a “nice young man,” but made no reference to Moore’s criminal and antisocial behavior before the murder. Even had some of these friends and relatives testified, we do not believe that, in light of the aggravating circumstances present in the case, there is a reasonable probability that the jury’s recommendation would have been different. Moreover, had such character testimony been introduced at the penalty hearing, the prosecution would have undoubtedly introduced its rebuttal evidence that would have given the jury an even dimmer view of the defendant. Nor can we say that, even if we were to assume that counsel failed to plead for Moore’s life, there is a reasonable probability that the result would have been different.
Moore also contends that counsel was ineffective because he failed to object to the trial court’s failure to define “especially heinous, atrocious, or cruel” to the jury, which Moore contends is required by State v. Sonnier, 402 So.2d 650, 658-59 (La.1981), cert. denied, — U.S. ---, 103 S.Ct. 3571, 77 L.Ed.2d 1412 (1983). As an initial matter, we note that in State v. Willie, 436 So.2d 553, 556 (La.1983), cert. denied, — U.S. ---, 104 S.Ct. 1327, 79 L.Ed.2d 723 (1984), the Louisiana Supreme Court held that “[wjhile Sonnier indicates that it is desirable for the trial court to instruct the jury about what constitutes a heinous crime, the square holding of that case does not mandate such an instruction.” More importantly, the United States Supreme Court has thus far declined to require that the jury must be instructed of the narrow construction of a potentially overbroad aggravating circumstance, such as whether the crime is “especially heinous, atrocious, or cruel.” Instead, the Court has looked to state appellate courts “to weed out those *319cases in which an overly broad construction is applied by the jury____” See Williams v. Maggio, 679 F.2d 381, 410 (5th Cir.1982) (en banc) (Randall, J., dissenting), cert. denied, — U.S. -, 103 S.Ct. 3553, 77 L.Ed.2d 1399 (1983). The Louisiana Supreme Court has held that for a murder to be “especially heinous,” there must be “ ‘evidence that there was torture or the pitiless infliction' of unnecessary pain on the victim.’ ” State v. Monroe, 397 So.2d 1258, 1275-76 (La.1981) (citation omitted), cert. denied, — U.S.-, 103 S.Ct. 3571, 77 L.Ed.2d 1411 (1983). Moore does not suggest that such did not exist in this case. Thus, Moore has not been prejudiced in this regard.
Moore next contends that counsel was ineffective because he made no objection to the failure of the trial judge to explain that if the jury was unable to agree unanimously on a verdict, the judge would be required to sentence the defendant to life imprisonment. The jury was informed, however, that the sentence it imposed, whether death or life imprisonment, had to be unanimous. Joint Exhibit 1C at 781-82. The jury was also instructed that:
If you find beyond a reasonable doubt that any of the statutory aggravating circumstances existed you may consider imposing a sentence of death. If, however, you do not unanimously find beyond a reasonable doubt that any of the statutory aggravating circumstances existed then life imprisonment without benefit of parole, probation or suspension of sentence is the only sentence that may be imposed.
Id. at 779. In Baldwin v. Blackburn, 653 F.2d 942, 952-53 (5th Cir.1981), cert. denied, 456 U.S. 950, 102 S.Ct. 2021, 72 L.Ed.2d 475 (1982), we held that these instructions sufficiently informed the jury that if even one member of the jury held out, the trial judge would be required to impose a life sentence. Thus, Moore was not prejudiced by counsel’s failure to object to the instructions given to the jury.9
Thus, because Moore has failed to show that any of counsel’s asserted errors were both unreasonable and prejudicial, he has failed to show that he received ineffective assistance of counsel at the penalty hearing.
III. ALTERNATIVE GROUNDS FOR HABEAS RELIEF.
Although the district court did not address Moore’s other claims of constitutional error alleged to have occurred at the sentencing trial, we conclude that none of them requires us to remand the case to the district court, since Moore is not entitled to partial habeas relief as a matter of law.
A. Meaningful Appellate Review.
Moore argues that the Louisiana Supreme Court failed to conduct the meaningful appellate review of his sentence that the United States Supreme Court has found to be a safeguard against arbitrary application of the death penalty.10 See *320Pulley v. Harris, — U.S. ---, 104 S.Ct. 871, 79 L.Ed.2d 29 (1984); California v. Ramos, — U.S. ---, 103 S.Ct. 3446, 77 L.Ed.2d 1171 (1983).11 First, Moore contends that the Louisiana Supreme Court’s determination that his sentence was not influenced by “passion, prejudice or other arbitrary factors” was “superficial at best.” In its review of Moore’s sentence, the court noted that “[a]t the sentencing hearing, the prosecutor advised the jury that ‘From the next point forward it goes to the court system to be thoroughly reviewed and checked through every court in this land.’ ” State v. Moore, 414 So.2d at 347. In previous cases, the Louisiana Supreme Court has vacated a defendant’s death sentence when it believed that the prosecution had induced the jury to believe that its responsibility was lessened by appellate review. See, e.g., State v. Willie, 410 So.2d 1019 (La.1982). See also State v. Monroe, 397 So.2d 1258 (La.1981) (discussing when prosecutor’s reference to appellate review requires defendant’s death sentence to be vacated), cert. denied, — U.S. ---, 103 S.Ct. 3571, 77 L.Ed.2d 1411 (1983); State v. Berry, 391 So.2d 406 (La.1980) (same), cert. denied, 451 U.S. 1010, 101 S.Ct. 2347, 68 L.Ed.2d 863 (1981). In the instant case, the court held that the prosecutor’s reference to appellate review, “although close to reversible error, did not induce the jury to believe that its responsibility was lessened by appellate review.” State v. Moore, 414 So.2d at 347 (citing State v. Mattheson, 407 So.2d 1150, 1165 (La.1981), cert. denied, — U.S. ---, 103 S.Ct. 3571, 77 L.Ed.2d 1412 (1983)). Simply because the court did not believe that Moore’s sentence was required to be vacated does not mean that he did not receive meaningful appellate review. See Proffitt v. Florida, 428 U.S. 242, 253, 258, 96 S.Ct. 2960, 2967, 2969, 49 L.Ed.2d 913 (1976) (opinion of Stewart, Powell, and Stevens, JJ.) (meaningful appellate review conducted where court’s review has been conducted with “maximum of rationality and consistency” and where court “has not hesitated to vacate a death sentence when it has determined that the sentence should not be imposed”). Moreover, we do not believe that the prosecutor’s brief reference to appellate review diminished the jury’s sense of responsibility for its sentence. See Corn v. Zant, 708 F.2d 549, 556-58 (11th Cir.1983), cert. denied, — U.S. ---, 104 S.Ct. 2670, 81 L.Ed.2d 375 (1984); McCorquodale v. Balkcom, 705 F.2d 1553, 1556 (11th Cir.1983), cert. denied, — U.S. ---, 104 S.Ct. 2161, 80 L.Ed.2d 546 (1984).
In considering whether the jury’s sentencing recommendation was influenced by prejudice, the Louisiana Supreme Court found it “significant that the three co-defendants were black and the victim was white, as were all the jurors.” State v. Moore, 414 So.2d at 347. The court noted, however, that “there is no reference at any point in the record to color, or appeal to racial prejudice. The crime was both violent and senseless; there is no indication that the verdict was influenced by prejudice.” Id. Despite the fact there is not a single reference to race in the record, Moore contends there was a risk that his death sentence was imposed because the victim was white and the defendant was black, and' that the Louisiana Supreme Court’s failure to recognize this rendered its review meaningless. This argument presumes that in every case in which the racial composition of victim and perpetrator differs, there is an inherent risk that the sentence results from racial prejudice. Such a presumption, without more, is untenable. Thus, we do not accept Moore’s position on this issue.
*321Finally, Moore contends that the Louisiana Supreme Court’s proportionality review was inadequate. However, subsequent to the date on which Moore’s habeas petition was filed, the United States Supreme Court decided in Pulley v. Harris, supra, that the eighth and fourteenth amendments do not require the states to conduct a proportionality review of the defendant’s sentence. 104 S.Ct. at 876.12
B. Unsupported Aggravating Circumstance.
Moore next contends that his death sentence was constitutionally infirm because there was insufficient evidence to support one of the aggravating circumstances found by the jury.13 In Williams v. Maggio, supra, we held that as long as one of the aggravating circumstances found by the jury is supported by the evidence, the defendant’s death sentence would not be overturned. 679 F.2d at 388-90. See Zant v. Stephens, 462 U.S. 862, 103 S.Ct. 2733, 77 L.Ed.2d 235 (1983).
C. Composition of Moore’s Jury.
Moore next argues that the process of excluding from juries persons who are unwilling to vote for capital punishment results in juries that are biased in favor of the prosecution and do not represent a fair cross section of the community.14 We rejected this very contention in Sonnier v. Maggio, 720 F.2d 401, 407 (5th Cir.1983), cert. denied, — U.S. ---, 104 S.Ct. 1331, 79 L.Ed.2d 726 (1984); see also Smith v. Balkcom, 660 F.2d 573, 575-84 (5th Cir.1981), cert. denied, 459 U.S. 882, 103 S.Ct. 181, 74 L.Ed.2d 148 (1982); Spinkellink v. Wainwright, 578 F.2d 582, 583-96 (5th Cir.1978), cert. denied, 440 U.S. 976, 99 S.Ct. 1548, 59 L.Ed.2d 796 (1979).
D. Jury Instructions.
Moore contends that the trial court erred in failing to define “especially heinous, atrocious, or cruel” to the jury. Moore also maintains that the court failed to explain to the jury that if it was unable to agree unanimously on a verdict, the judge would be required to sentence Moore to life imprisonment.15 We have explained why these instructions are not constitutionally infirm in part H.D., supra.
E. Discriminatory Application of the Death Penalty.
Finally, Moore contends that he “was sentenced to death pursuant to a *322pattern and practice of racially discriminatory and arbitrary infliction of the death penalty” in contravention of the eighth and fourteenth amendments.16 He alleges that “[t]he death penalty in the United States and in the State of Louisiana has been discriminatorily imposed against blacks and blacks accused of killing whites.”
Moore’s conc'lusory allegations do not entitle him to relief. See, e.g., Spinkellink v. Wainwright, supra, at 614 n. 40. Moreover, in light of the evidence Moore proffered to the district court, we do not believe that an evidentiary hearing is required on this issue. As in Smith v. Balkcom, 660 F.2d 573 (5th Cir.1981), modified, 671 F.2d 858 (5th Cir.), cert. denied, 459 U.S. 882, 103 S.Ct. 181, 74 L.Ed.2d 148 (1982), the statistics proffered in the instant case are incomplete. The tables do not take into account the various statutory aggravating circumstances such as the “especially heinous, atrocious, or cruel” murder evidence in this case. See also McCorquodale v. Balkcom, 705 F.2d at 1556 (no evidentiary hearing required where “tables do not take into account the various aggravating circumstances such as the ‘wantonly vile, horrible [and] inhumane’ torture-murder evidence here”).
IV. CONCLUSION.
We conclude that Moore was not denied effective assistance of counsel at the sentencing trial. Nor has Moore asserted any other grounds that require him to have a new penalty hearing. We AFFIRM that part of the district court’s judgment that denies Moore habeas relief as to his conviction. We REVERSE that part of the judgment that vacates Moore’s death sentence and orders a new penalty hearing. The district court’s stay of execution is VACATED.
. The facts of the case are summarized from State v. Moore, 414 So.2d 340 (La.1982).
. See generally La.Code Crim.Proc. art. 905 et seq. (West Supp.1983).
. The court held:
In finding that Moore created a risk of death or great bodily harm to more than one person, the jurors were apparently thinking of baby Regina. Although the murder of her mother certainly placed the child in an unprotected situation, she was not harmed or threatened. The evidence does not support this aggravating circumstance.
State v. Moore, 414 So.2d at 348. The court found that there was sufficient evidence to support the jury’s finding of the other two aggravating circumstances, and thus Moore’s death sentence was left intact. See, e.g., State v. Monroe, 397 So.2d 1258, 1275-76 (La.1981), cert. denied, — U.S. -, 103 S.Ct. 3571, 77 L.Ed.2d 1411 (1983).
. The district court granted Moore’s motion for a certificate of probable cause to cross-appeal.
. Moore’s petition for habeas corpus relief alleges:
At petitioner’s sentencing hearing at which the death penalty was imposed, petitioner was deprived of his right to the effective assistance of counsel guaranteed by the Sixth Amendment to the Constitution of the United States____ Defense counsel did not interview known potential witnesses for petitioner at the sentencing hearing or conduct any presen-
tencing hearing investigation. At the sentencing hearing itself, defense counsel did not offer any evidence or plead for petitioner's life. Defense counsel’s failures to investigate and develop defensive mitigating evidence were not the result of strategic choices, and were not based on reasonable assumptions about the proper conduct of petitioner’s defense. Defense counsel’s conduct constitutes pervasive ineffective assistance of counsel causing actual and substantial disadvantage to petitioner’s fate.
. In Strickland v. Washington, the Court noted that ‘‘[i]n preparing for the sentencing hearing, counsel spoke with respondent about his background. He also spoke on the telephone with respondent’s wife and mother, though he did not follow up on the one successful effort to meet them. He did not otherwise seek out character witnesses for respondent." 104 S.Ct. at 2057. The Court held that counsel’s decision not to seek other possible mitigation witnesses was reasonable because "[tjrial counsel could reasonably surmise from his conversations with respondent that character and psychological evidence would be of little help.” Id. at 2071.
. Because counsel’s voir dire was not infirm, we, of course, do not need to consider whether Moore was prejudiced in this regard. See infra note 9.
. Moore also argues that counsel was unprepared for the sentencing phase by referring to certain events that transpired at the guilt phase. The district court found, however, that Moore had effective representation at the guilt phase and Moore does not appeal that determination.
. We recognize that we have only examined Moore’s allegation that counsel was ineffective for not objecting to the court’s instructions under the prejudice prong of Strickland v. Washington. We need not address whether counsel’s performance was deficient if the defendant has failed to meet the prejudice prong. See Strickland v. Washington, 104 S.Ct. at 2069.
. Moore’s petition alleges:
Petitioner’s death sentence was unconstitutionally imposed in violation of the Eighth and Fourteenth Amendments to the Constitution of the United States as the result of the Louisiana Supreme Court’s failure to engage in meaningful appellate review designed to assure that death is the appropriate sentence. In reviewing petitioner’s death sentence, the
Louisiana Supreme Court failed to consider adequately that the death penalty was imposed as a result of passion, prejudice and other arbitrary factors, including racial prejudice and the injection of appellate review. The Louisiana Supreme Court also failed to consider that the sentencing jury’s consideration of [an unsupported] statutory aggravating factor skewed the balance struck by the jury in deciding upon death. The Louisiana Supreme Court’s proportionality review itself evidenced that the jury’s decision to vote for death was the result of passion, prejudice or other arbitrary factors. The Louisiana Supreme Court’s conclusion to the contrary evidences its failure to conduct a meaningful appellate review.
. Louisiana Supreme Court Rule 28 provides:
Every sentence of death shall be reviewed by this court to determine if it is excessive. In determining whether the sentence is excessive the court shall determine:
(a) whether the sentence was imposed under the influence of passion, prejudice or any other arbitrary factors, and
(b) whether the evidence supports the jury's finding of a statutory aggravating circumstance, and
(c) whether the sentence is disproportionate to the penalty imposed in similar cases, considering both the crime and the defendant.
Codified at La.Code Crim.Proc. 905.9.1(1) (West Supp.1983).
. Moore also contends that he failed to receive meaningful appellate review because the Louisiana Supreme Court failed to consider whether the jury’s verdict would have been different absent the unsupported aggravating circumstances. We rejected this contention in Williams v. Maggio, supra. See infra part III.B.
. Moore's petition alleges:
Petitioner was sentenced to death in violation of the Eighth and Fourteenth Amendments to the Constitution of the United States because one of the three aggravating circumstances found by the jury was subsequently held by the Louisiana Supreme Court to be unsupported by the record evidence. The jury’s consideration of a statutory aggravating factor — that petitioner created the risk of danger or great bodily harm to more than one person — unsupported by the evidence impermissibly skewed the balance struck by the jury in voting for death and injected an element of unreliability in the death-determination that is constitutionally unacceptable.
. Moore’s petition alleges:
Petitioner was convicted and sentenced to death in violation of the Sixth, Eighth and Fourteenth Amendments to the Constitution of the United States as the result of the prosecution’s exclusion from the jury all prospective jurors opposed to the death penalty.
. Moore’s petition alleges:
Petitioner was deprived of his right to the guided exercise of the sentencer’s discretion in a capital case in violation of the Sixth, Eighth and Fourteenth Amendments to the Constitution of the United States as a result of the trial court’s failure to define "especially heinous, atrocious, or cruel,” La.C.Cr.P. Art. 905.4(g), and to explain to the jury that if they were unable to agree unanimously on a verdict, the judge would be required by Louisiana law to sentence the defendant to life imprisonment, La.C.Cr.P. Art. 905.8.
. Moore’s petition alleges:
Petitioner was sentenced to death in violation of the Eighth and Fourteenth Amendments to the Constitution of the United States as a result of the racially discriminatory and arbitrary manner in which the death penalty was administered.