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Polya-Schoenberg Conjecture Type Problems For Harmonic Univalent Mappings

Polya-Schoenberg Conjecture Type Problems For Harmonic Univalent Mappings. By Om P. Ahuja, Jay M. Jahangiri Kent State University, Ohio H. Silverman, College of Charleston, SC. Presented at the Joint Annual Conference: AMS – MAA, Baltimore January 15, 2003. y. v. f(z). z. x. 0. u.

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Polya-Schoenberg Conjecture Type Problems For Harmonic Univalent Mappings

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  1. Polya-Schoenberg Conjecture Type Problems For Harmonic Univalent Mappings By Om P. Ahuja, Jay M. Jahangiri Kent State University, Ohio H. Silverman, College of Charleston, SC Presented at the Joint Annual Conference: AMS – MAA, Baltimore January 15, 2003

  2. y v f(z) z x 0 u 0 Harmonic function

  3. y f v f(z) z x 0 0 u Definitions

  4. Basic Facts

  5. Def: Convolution of harmonic functions

  6. Polya-Schoenberg Conjecture Theorem A [Ruscheweyh and Sheil-Small, 1973] This is the famous Polya-Schoenberg conjecture (1958). We obtain a modification of Theorem A for close-to-convex harmonic case. Theorem 1 Then

  7. Lemma 1 (Clunie and Sheil-Small, 1984)

  8. Lemma 2 [Ruscheweyh and Sheil-Small, 1973]

  9. Proof of Theorem 1

  10. Proof of Theorem 1 Continued This completes the proof. Corollary 1:

  11. Theorem B (Ruscheweyh and Sheil-Small, 1973) We extend Theorem B to close-to-convex harmonic case. Theorem 2:

  12. Proof of Theorem 2

  13. Corollary 2 Proof.

  14. Theorem 3

  15. Theorem 4

  16. Thank You

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