[PUBLISH]
Before PRYOR, MARTIN and EDMONDSON, Circuit Judges.
Anthony Ponticelli, a Florida prisoner sentenced to death for the murder of two brothers, Nick and Ralph Grandinetti, raises two issues about the denial of his petition for a writ of habeas corpus. First, Ponticelli argues that the prosecution violated his right to due process when it allegedly suppressed evidence of and failed to correct false testimony about an agreement to provide immunity for a witness for the state and about Ponticelli’s use of cocaine shortly before the murders, see Brady v. Maryland, 373 U.S. 83, 83 S. Ct. 1194 (1963); see Giglio v. United States, 405 U.S. 150, 92 S. Ct. 763 (1972). Ponticelli contends that the ruling of the Supreme Court of Florida-that the prosecution did not violate his due process rights and that, even if it did, he suffered no prejudice-is contrary to or an unreasonable application of clearly established federal law and an unreasonable determination of the facts. 28 U.S.C. §2254(d). Second, Ponticelli argues that his trial counsel provided ineffective assistance by failing to present evidence of Ponticelli’s incompetence to stand trial and by failing to present mitigating evidence of drug use and mental health problems during the penalty phase. See Strickland v. Washington, 466 U.S. 668, 104 S. Ct. 2052 (1984). Ponticelli contends that the ruling of the Supreme Court of Florida-that trial counsel did not render deficient performance before and during the competency hearing, and that any deficiencies by counsel during the penalty phase did not prejudice Ponticelli-is contrary to or an unreasonable application of clearly established federal law. 28 U.S.C. § 2254(d). Ponticelli’s arguments fail. The Supreme Court of Florida on the first issue, reasonably determined the underlying facts and, on both issues, neither contravened nor unreasonably applied clearly established federal law. The denial of Ponticelli’s petition for a writ of habeas corpus is affirmed.