On functions \(\alpha\)-starlike with respect to symmetric conjugate points
Let \(A\) denote the class of functions \(f(z) = z + a_2 z^2 + \cdots\) regular in \(U = \{z : |z |<1\}\) and let \[ Tf(z) : = \bigl( f(z) - \overline {f (-\overline z)} \bigr)/2 \] \[ D^0 f(z) : = f(z),\;D^1f(z) = Df(z) : = zf'(z), \] \[ D^{n + 1} f(z) : = D \bigl( D^n f(z) \bigr), \quad n = 1, 2, \dots \] A function \(f \in A\) is said to be \(\alpha\)-starlike with respect to symmetric conjugate points, if \[ \text{Re} {D \bigl( \alpha D + (1 - \alpha) D^0 \bigr) f(z) \over \bigl( \alpha D + (1 - \alpha) D^0 \bigr) Tf(z)} > 0, \quad z \in U, \] for some \(\alpha \geq 0\). In this paper some properties of this class as an interesting convolution theorem, coefficient estimates and a structural formula are obtained. In particular it is proved that if a function \(f(z) \in A\) is \(\alpha\)-starlike with respect to symmetric points then \(|a_n |\leq {n \over 1 - (n - 1) n}\), \(n = 2, 3, \dots\). The estimate is sharp and the equality is attained for the function \(g_\alpha (z) = z + \sum^\infty_{k = 2} {k (i)^{k - 1} \over 1 + (n - 1) k} z^k\).
starlike with respect to symmetric points
scientific article; zbMATH DE number 908614