论文标题
在精确差分从属的凸优势上
On Convex Dominants of Exact Differential Subordination
论文作者
论文摘要
令$ h $为非消失的凸函数，$ p $为$ \ mathbb {d} $中的分析函数。我们考虑差异从属$$ψ_i（p（z），z p'（z））\ prec h（z）$$，可容纳的功能为$ψ_1：=（βp（z）+γ）^{ - α} { - α} { - α} \ lesg p'（z）\右）$和$ψ_2：= \ tfrac {1} {\ sqrt {\ sqrt {\ sqrt {\ arctan \ left（\ sqrt {\tfracβγγ} p^{1-α}（1-α}（z）（z）\右） （1-α）}（z）+γ} \ right）\ tfrac {z p'（z）} {p^α（z）} $。本文的目的是找到优势，最好是满足$ψ_i（q，n zq'（z））= h（z）$的上述差分下属解决方案的最佳优势（例如$ q $）。此外，我们表明$ψ_i（q，zq'（z））= h（z）$是一个精确的微分方程，$ q $是$ \ mathbb {d} $中的convex Univalent函数。此外，我们估计$ h $的不同选择的$ \ re p $的尖锐下限，并得出了$ \ Mathcal {h} $（分析归一化功能类）功能的单相标准，以此作为我们的结果。
Let $h$ be a non vanishing convex univalent function and $p$ be an analytic function in $\mathbb{D}$. We consider the differential subordination $$ψ_i(p(z), z p'(z)) \prec h(z)$$ with the admissible functions in consideration as $ψ_1:=(βp(z)+γ)^{-α}\left(\tfrac{(βp(z)+γ)}{β(1-α)}+ z p'(z)\right)$ and $ψ_2:=\tfrac{1}{\sqrt{γβ}}\arctan\left(\sqrt{\tfracβγ}p^{1-α}(z)\right)+\left(\tfrac{1-α}{βp^{2 (1-α)}(z)+γ}\right)\tfrac{z p'(z)}{p^α(z)}$. The objective of this paper is to find the dominants, preferably the best dominant(say $q$) of the solution of the above differential subordination satisfying $ψ_i(q, n zq'(z))= h(z)$. Further, we show that $ψ_i(q,zq'(z))= h(z)$ is an exact differential equation and $q$ is a convex univalent function in $\mathbb{D}$. In addition, we estimate the sharp lower bound of $\RE p$ for different choices of $h$ and derive a univalence criteria for functions in $\mathcal{H}$(class of analytic normalized functions) as an application to our results.
