Mapping Radius of Regular Function and Center of Convex Region

Abstract

There is only a univalent and regular function in a simply connected region. It transforms the region into a unit circle, while satisfies that the function value of a point is equal to zero and the derivative of this point is greater than zero, then the inverse of the derivative is named as the mapping radius of the function at the point. The mapping radius is changing according to the movement of the point. In fact, the mapping radius in the region is a real valued function. The function is continuous and reaches the maximum value within the region. If a simply connected region is the convex region with the symmetrical center, then the mapping radius function of the region obtains the maximum at the symmetrical center of the region. Introduction First of all, we look back on the famous Riemann’ Theorem and his deduction[1]. Riemann’ Theorem: Supposing the boundary points of the simply connected region G on the plane Z are not limited to one, 0 z G ∈ , there is the only univalent and regular function ( ) f z that G will be transformed into the unit circle: 1 w . Deduction: Supposing the boundary points of the simply connected region G on the plane Z are not limited to one, 0 z G ∈ , there is the only univalent and regular function ( ) F z that G will be transformed to the unit circle: 0 1 ( ) w f z . Its inverse function is marked as ( ) z w φ = . Again mark ( ( )) ( ) ( ) F w w M F φ ψ = , then ( ) w ψ is the univalent and regular function on the unit circle: 1 w < , (0) 0 ψ = , ( ) 1 w ψ < , according to Schwarz’ Theorem, (0) 1 ψ ′ ≤ . because ( ( )) ( ( )) 1 ( ) ( ) ( ) ( ) ( ) F w F w w w M F M F f z φ φ ψ φ ′ ′ ′ ′ = = ′ , 0 0 ( (0)) 1 1 (0) ( ) ( ) ( ) ( ) F M F f z M F f z φ ψ ′ ′ = = ′ ′ ,