Differential Inequalities and Univalent Functions

Abstract

Let ${\mathcal M}$ be the class of analytic functions in the unit disk $\ID$ with the normalization $f(0)=f'(0)-1=0$, and satisfying the condition $$\left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq 1, \quad z\in \ID. $$ Functions in $\mathcal{M}$ are known to be univalent in $\ID$. In this paper, it is shown that the harmonic mean of two functions in ${\mathcal M}$ are closed, that is, it belongs again to ${\mathcal M}$. This result also holds for ot

Results & Lemmas (3)

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Theorem 1. Theorem 1. Let f, g ∈U satisfy f(z)+g(z) z ̸= 0 for z ∈D. Then the function F given by (1) also belongs to the class U.

Theorem 1. Let f, g ∈U satisfy f(z)+g(z) z ̸= 0 for z ∈D. Then the function F given by (1) also belongs to the class U.

Theorem 2. Theorem 2. Suppose f, g ∈M satisfy f(z)+g(z) z ̸= 0 for z ∈D. Then the function F given by (1) also belongs to the class M.

Theorem 2. Suppose f, g ∈M satisfy f(z)+g(z) z ̸= 0 for z ∈D. Then the function F given by (1) also belongs to the class M.

Theorem 3. Theorem 3. Let f, g ∈N satisfy f(z)+g(z) z ̸= 0 for z ∈D. Then the function F given by (1) also belongs to the class N.

Theorem 3. Let f, g ∈N satisfy f(z)+g(z) z ̸= 0 for z ∈D. Then the function F given by (1) also belongs to the class N .

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