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Extremal Problems for Hyperbolically Convex Functions

This paper discusses three significant extremal problems related to hyperbolically convex functions. The authors, Roger W. Barnard, Kent Pearce, and G. Brock Williams, delve into the concepts of hyperbolic geodesics and hyperbolically convex sets. The work examines various theorems and provides proofs relevant to the calculus of variations, including discussions on Julia variations. This exploration contributes to the understanding of the properties and applications of hyperbolically convex functions within computational methods and function theory.

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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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