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Three Extremal Problems for Hyperbolically Convex Functions

Three Extremal Problems for Hyperbolically Convex Functions. Roger W. Barnard, Kent Pearce, G. Brock Williams Texas Tech University [Computational Methods and Function Theory 4 (2004) pp 97-109]. Notation & Definitions. Notation & Definitions. Notation & Definitions.

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Three Extremal Problems for Hyperbolically Convex Functions

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  1. Three Extremal Problems for Hyperbolically Convex Functions Roger W. Barnard, Kent Pearce, G. Brock Williams Texas Tech University [Computational Methods and Function Theory 4 (2004) pp 97-109]

  2. Notation & Definitions

  3. Notation & Definitions

  4. Notation & Definitions • Hyberbolic Geodesics

  5. Notation & Definitions • Hyberbolic Geodesics • Hyberbolically Convex Set

  6. Notation & Definitions • Hyberbolic Geodesics • Hyberbolically Convex Set • Hyberbolically Convex Function

  7. Notation & Definitions • Hyberbolic Geodesics • Hyberbolically Convex Set • Hyberbolically Convex Function • Hyberbolic Polygon o Proper Sides

  8. Classes

  9. Classes

  10. Classes

  11. Classes

  12. Examples

  13. Problems • 1.

  14. Problems • 1. • 2. Find

  15. Problems • 1. • 2. Find • 3.

  16. Theorem 1

  17. Theorem 2 Remark Minda & Ma observed that cannot be extremal for

  18. Theorem 3

  19. Julia Variation

  20. Julia Variation (cont.)

  21. Julia Variation (cont.)

  22. Variations for (Var. #1)

  23. Variations for (Var. #2)

  24. Proof (Theorem 1)

  25. Proof (Theorem 1)

  26. Proof (Theorem 1)

  27. Proof (Theorem 1) From the Calculus of Variations:

  28. Proof (Theorem 1)

  29. Proof (Theorem 1)

  30. Proof (Theorem 1)

  31. Proof (Theorem 1)

  32. Proof (Theorem 1)

  33. Proof (Theorem 1)

  34. Proof (Theorem 1)

  35. Proof (Theorem 1)

  36. Proof (Theorem 1)

  37. Proof (Theorem 1)

  38. Proof (Theorem 1)

  39. Proofs (Theorem 2 & 3)

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