Generalized Kaplan classes and their applications

Szymon Ignaciuk, Maciej Parol

Abstract


Ali and Vasudevarao considered the integral operator \(I_{r,s}(z):=\int_{0}^{z}(f'(t))^r(g'(t))^s d t\)  and determined all values of \(r\) and $s$ for which the operator \((f,g)\mapsto I_{r,s}\) maps a specified subclass of Hornich space into another specified subclass of Hornich space. Thus, as it was stated by Kumar and Sahoo, Ali and Vasudevarao studied the range of \(r\) and \(s\) that preserves properties of these specified classes. Based on the Kaplan classes, we introduce the product classes \(K_{a,b}\) for arbitrary finite sequences \(a\) and \(b\) and consider operations similar to Hornich operations.  To this end we improve Sheil-Small's factorization theorem. Moreover, using elaborated techniques, we simplify proofs and solve the generalized problems considered by Causey and Reade, Goodman, Kim and Merkes.

Keywords


Kaplan classes; univalence; integral operators; convex functions; starlike functions; close-to-convex functions

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References


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DOI: http://dx.doi.org/10.17951/a.2025.79.1.53-74
Date of publication: 2025-07-31 20:53:34
Date of submission: 2025-07-23 23:25:47


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