Three Extremal Problems for Hyperbolically Convex Functions

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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

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Notation Definitions

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Notation Definitions

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Notation Definitions

• Hyberbolic Geodesics

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Notation Definitions

• Hyberbolic Geodesics
• Hyberbolically Convex Set

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Notation Definitions

• Hyberbolic Geodesics
• Hyberbolically Convex Set
• Hyberbolically Convex Function

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Notation Definitions

• Hyberbolic Geodesics
• Hyberbolically Convex Set
• Hyberbolically Convex Function
• Hyberbolic Polygon
• o Proper Sides

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Classes

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Classes

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Classes

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Classes

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Examples

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Problems

• 1.

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Problems

• 1.
• 2.
• Find

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Problems

• 1.
• 2.
• Find
• 3.

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Theorem 1

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Theorem 2

• Remark Minda Ma observed that cannot be
  extremal
• for

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Theorem 3

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Julia Variation

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Julia Variation (cont.)

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Julia Variation (cont.)

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Variations for (Var. 1)

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Variations for (Var. 2)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

• From the Calculus of Variations

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proof (Theorem 1)

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Proofs (Theorem 2 3)