If A is the class of all analytic functions in the complex unit disc $\Delta$, of the form: $f(z) = z + a_2z^2 + \cdots$ and if $f \in A$ satisfies in $\Delta$ the condition: $$Re \frac{zf'(z)}{f(z)} > |\frac{zf'(z)}{f(z)}-1|$$ then Re \sqrt[n]{f(z)/z \geq (n+1)/(n+2)}. We show also that if $f$ is starlike in $\Delta$ (i.e. Re zf'(z)/f(z) > 0 in $\Delta$), then Re \sqrt[n]{f(z)/z > n(n+2)}.
PETTINEO M (2004). Inequalities concerning starlike functions and their n-th root. GENERAL MATHEMATICS, 12, 49-56.
Inequalities concerning starlike functions and their n-th root
PETTINEO, Maria
2004-01-01
Abstract
If A is the class of all analytic functions in the complex unit disc $\Delta$, of the form: $f(z) = z + a_2z^2 + \cdots$ and if $f \in A$ satisfies in $\Delta$ the condition: $$Re \frac{zf'(z)}{f(z)} > |\frac{zf'(z)}{f(z)}-1|$$ then Re \sqrt[n]{f(z)/z \geq (n+1)/(n+2)}. We show also that if $f$ is starlike in $\Delta$ (i.e. Re zf'(z)/f(z) > 0 in $\Delta$), then Re \sqrt[n]{f(z)/z > n(n+2)}.
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