New classification of analytic functions with negative coefficients
Let \(A_{p,n}\) denote the class of functions of the form \[ f(z)=z\quad p-\sum^{\infty}_{k=p+n}a_ kz\quad k,\quad a_ k\geq 0, \] p, \(n\in \{1,2,3,...\}\), which are analytic in the unit disc \textit{U. H. Silvermann} [Proc. Am. Math. Soc. 51, 109-116 (1975; Zbl 0311.30007)] has obtained necessary and sufficient conditions in terms of \(\{a_ n\}\) for the class of \(A_{1,n}\) and many authors have obtained such conditions for subclasses of univalent or p-valent analytic functions with negative coefficients. In this paper, the authors unify all the results by defining the class \(A(p,n,B_ k)\) of all functions of \(A_{p,n}\), which satisfy the inequality \(\sum^{\infty}_{k=p+n}B_ ka_ k\leq 1\). Various distortion theorems for these functions and for fractional calculus of functions belonging to \(A(p,n,B_ k)\) are given and some consequences are discussed.
scientific article; zbMATH DE number 4032032