WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 23, 2024
Estimates and Radii of Convexity in Some Classes of Regular Functions
Authors: , , , ,
Abstract: A class $$C_{n} (λ,δ,α,γ)$$ is being introduced regular in the circle $$Ε=\lbrace z:|z|<1\rbrace$$ functions $$f(z)$$, satisfying the condition $$|((1-λz^{n})(1-δz^{n})f'(z))^{\frac{1}{γ}}-α|\leqslantα,z\in E$$, where $$λ,δ\geqslant 0,0<γ\leqslant 1,α>\frac{1}{2},n \geqslant 1$$. Class $$C_{n} (λ,δ,α,γ)$$ generalizes various subclasses of close-to-convex functions, including functions which are convex in a certain direction and functions with limited rotation. Estimates of the derivative and logarithmic derivative of the function $$f (z) \in C_{n} (λ,δ,α,γ)$$ are found, and also the radii of the convexity of the class $$C_{n} (λ,δ,α,γ)$$. The case is also considered when the function $$f(z)$$ has gaps in the expansion in a row. Similar results are formulated for the class $$T_{n} (λ,δ,α,γ)$$ of functions $$F(z)$$, satisfying the condition $$|((1-λz^{n})(1-δz^{n})f'(z))^{\frac{1}{γ}}-α|\leqslantα,z\in E$$, which generalizes classes of typically real and close-to-starlike functions. All results are accurate. With the appropriate selection of parameter values of $$λ,δ,α,γ,n$$ both new and previously published results are obtained.
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Keywords: geometric theory of functions, estimates of regular functions, radii of convexity, close-to-convex functions, typically real functions, close-to-starlike functions
Pages: 446-457
DOI: 10.37394/23206.2024.23.47