Three Extremal Problems for Hyperbolically Convex Functions Roger W. Barnard, Kent Pearce, G. Brock Williams Texas Tech University [Computational Methods.

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Presentation transcript:

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 ]

Notation & Definitions

Notation & Definitions

Notation & Definitions Hyberbolic Geodesics

Notation & Definitions Hyberbolic Geodesics Hyberbolically Convex Set

Notation & Definitions Hyberbolic Geodesics Hyberbolically Convex Set Hyberbolically Convex Function

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

Classes

Classes

Classes

Classes

Examples

Problems 1.

Problems Find

Problems Find 3.

Theorem 1

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

Theorem 3

Julia Variation

Julia Variation (cont.)

Variations for (Var. #1)

Variations for (Var. #2)

Proof (Theorem 1)

From the Calculus of Variations:

Proof (Theorem 1)

Proofs (Theorem 2 & 3)