On the order of convolution consistence of the analytic functions with negative coefficients

Abstract

Making use of a modified Hadamard product, or convolution, of analytic functions with negative coefficients, combined with an integral operator, we study when a given analytic function is in a given class. Following an idea of U. Bednarz and J. Sokół, we define the order of convolution consistence of three classes of functions and determine a given analytic function for certain classes of analytic functions with negative coefficients.

Keywords

• analytic function with negative coefficients
• univalent function
• extreme point
• order of convolution consistence
• starlikeness
• convexity