As special cases, the properties of $p$−valent starlikeness and $p$−valent convexity of the operators $\int_{0}^{z} pt^{p-1} \left( \frac{f(t)}{t^p}\right)^\delta dt$ and $\int_{0}^{z} pt^{p-1} \left( \frac{g'(t)}{pt^{p-1}}\right)^\delta dt$ are given.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we define two new general $p$−valent integral operators in the unit disc $U$ and obtain the properties of $p$−valent starlikeness and $p$−valent convexity of these integral operators of $p$−valent functions on some classes of $\beta$−uniformly $p$−valent starlike and $\beta$−uniformly $p$−valent convex functions of complex order and type α $(0 ≤\leq \alpha < p)$.
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Key Findings & Data:In this paper, we define two new general $p$−valent integral operators in the unit disc $U$ and obtain the properties of $p$−valent starlikeness and $p$−valent convexity of these integral operators of $p$−valent functions on some classes of $\beta$−uniformly $p$−valent starlike and $\beta$−uniformly $p$−valent convex functions of complex order and type α $(0 ≤\leq \alpha < p)$.
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Conclusion & Impact:As special cases, the properties of $p$−valent starlikeness and $p$−valent convexity of the operators $\int_{0}^{z} pt^{p-1} \left( \frac{f(t)}{t^p}\right)^\delta dt$ and $\int_{0}^{z} pt^{p-1} \left( \frac{g'(t)}{pt^{p-1}}\right)^\delta dt$ are given.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Analytic functions · Integral operators · $\beta$−uniformly $p$−valent starlike and $\beta$−uniformly $p$−valent convex functions
Abstract
In this paper, we define two new general $p$−valent integral operators in the unit disc $U$ and obtain the properties of $p$−valent starlikeness and $p$−valent convexity of these integral operators of $p$−valent functions on some classes of $\beta$−uniformly $p$−valent starlike and $\beta$−uniformly $p$−valent convex functions of complex order and type α $(0 ≤\leq \alpha < p)$. As special cases, the properties of $p$−valent starlikeness and $p$−valent convexity of the operators $\int_{0}^{z} pt^{p-1} \left( \frac{f(t)}{t^p}\right)^\delta dt$ and $\int_{0}^{z} pt^{p-1} \left( \frac{g'(t)}{pt^{p-1}}\right)^\delta dt$ are given.
DENİZ, E., DENİZ, E., MUSTAFA, N. (2022). Some starlikeness and convexity properties for two new $p$−valent integral operators. Hacettepe Journal of Mathematics and Statistics, Cilt: 45 - Sayı: 5, 1367–1378. https://ajindex.com/articles/530227