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Univalence criterion and convexity for an integral operator. (English) Zbl 1251.30029

Let \(f\) and \(g\) be normalized analytic functions defined on the open unit disk of the complex plane satisfying the conditions \[ |z^2f'(z)/f^2(z) -1|<1,\quad |f''(z)/f'(z)|<1\quad\text{and}\quad |g(z)|<M \] for some \(M>0\). For a complex number \(\alpha\) with non-negative real part satisfying \(2\sqrt{3}(1+2M^2)|\alpha|\leq 9\), the authors show that the integral \[ F(z):=\int_0^z[f'(t)\exp(g(t))]^\alpha dt \] is univalent in the unit disk. They also determine the order of convexity of the function \(F\) when \(f\) and \(g\) satisfy the conditions \[ |f'(z) (z/f(z))^\mu -1|<1-\beta,\quad |f''(z)/f'(z)|<1\quad\text{and}\quad |g(z)|<M. \]

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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References:

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