The Sharp Estimation of Functional in The Generalized Caratheodory Class

Abstract

This paper is concernrd with the precise determination of certain linear functional defiend on the generalized Caratheodory class, i.e the class of of analytic functions in the unit disc  that admit a Riemann-Stieltijes integral representation of the form:   with a kernel   where:  is a non-decreasing function on the interval  and satisfes the condition   and      are some arbitrary on the interval  , This class is denoted by  

The main focus of the investigation is the determination of the range of variability of the functional ,   fixed but arbitrary in the unit disk.

It has been  shown that the convex hull of the domain of the range of variability of the functional is generated by the derivative of the kernel at the point  

In addition, the estimation for lower and upper bounds of  are established  by By proving the following inequalities

for

The sharpness of the inequalities holds. The extremal function is                                       ,the equalities holds from wright when  and from left when  .