It is known that if f⁢(z)=zp+∑n=p+1∞an⁢zn{f(z)=z^{p}+\sum_{n=p+1}^{\infty}a_{n}z^{n}} and it is analytic in a convex domain D⊂ℂ{D\subset\mathbb{C}} and, for some real α, we have ℜ⁢𝔢⁡{exp⁡(i⁢α)⁢f(p)⁢(z)}>0{\operatorname{\mathfrak{Re}}\{\exp(i\alpha)f^{(p)}(z)\}>0}, z∈D{z\in D}, then f⁢(z){f(z)} is at most p -valent in D . This Ozaki condition is a generalization of the well-known Noshiro–Warschawski univalence condition. In this paper, we consider the radius of univalence of functions g⁢(z)=z+∑n=1∞bn⁢zn{g(z)=z+\sum_{n=1}^{\infty}b_{n}z^{n}} such that g′⁢(z)≺[(1+z)2/(1-z)2]{g^{\prime}(z)\prec[(1+z)^{2}/(1-z)^{2}]} and some related problems.

中文翻译：

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关于尾崎病情的扩展

已知如果f⁢（z）= zp + ∑n = p +1∞an⁢zn{f（z）= z ^ {p} + \ sum_ {n = p + 1} ^ {\ infty} a_ { n} z ^ {n}}并在凸域D⊂ℂ{D \ subset \ mathbb {C}}}中进行解析，对于某些实数α，我们有ℜ⁢𝔢⁡{exp⁡（i⁢α） ⁢f（p）⁢（z）}> 0 {\ operatorname {\ mathfrak {Re}} \ {\ exp（i \ alpha）f ^ {（p）}（z）\}> 0}，z∈D {z \ in D}，则f⁢（z）{f（z）}在D中至多为p价。这个Ozaki条件是众所周知的Noshiro-Warschawski一元性条件的推广。在本文中，我们考虑函数g =（z）= z + ∑n =1∞bn⁢zn{g（z）= z + \ sum_ {n = 1} ^ {\ infty} b_ {n}的单调半径z ^ {n}}，使得g′⁢（z）≺[（1 + z）2 /（1-z）2] {g ^ {\ prime}（z）\ prec [（1 + z）^ { 2} /（1-z）^ {2}]}和一些相关问题。