Open AccessConvolution conditions for bounded \(\alpha\)-starlike functions of complex order
Author(s) -
A. Y. Lashin
Publication year - 2017
Publication title -
annales universitatis mariae curie-skłodowska. sectio a, mathematica
Language(s) - English
Resource type - Journals
eISSN - 2083-7402
pISSN - 0365-1029
DOI - 10.17951/a.2017.71.1.65
Subject(s) - bounded function , convolution (computer science) , normalization (sociology) , mathematics , combinatorics , prime (order theory) , order (exchange) , complex plane , class (philosophy) , alpha (finance) , univalent function , regular polygon , pure mathematics , analytic function , mathematical analysis , computer science , geometry , construct validity , statistics , finance , machine learning , artificial intelligence , sociology , artificial neural network , anthropology , economics , psychometrics
Let \(A\) be the class of analytic functions in the unit disc \(U\) of the complex plane \(\mathbb{C}\) with the normalization \(f(0)=f^{^{\prime }}(0)-1=0\). We introduce a subclass \(S_{M}^{\ast }(\alpha ,b)\) of \(A\), which unifies the classes of bounded starlike and convex functions of complex order. Making use of Salagean operator, a more general class \(S_{M}^{\ast }(n,\alpha ,b)\) (\(n\geq 0\)) related to \(S_{M}^{\ast }(\alpha ,b)\) is also considered under the same conditions. Among other things, we find convolution conditions for a function \(f\in A\) to belong to the class \(S_{M}^{\ast }(\alpha ,b)\). Several properties of the class \(S_{M}^{\ast }(n,\alpha ,b)\) are investigated.