Radius of convexity for certain multivalent functions with missing coefficients (Q1932164)

Let \(U\) be the complex unit disc and \(A_p(n)\) the class of functions \(f\) with \(f(z)=z^p+\sum_{k=n}^\infty a_{p+k}z^{p+k}\), \(n,p\in\mathbb N^*\), analytic and \(p\)-valent in \(U\). The author investigates radius properties of such analytic \(p\)-valent functions. The principal result is the following one, and, it is also sharp: If \(f\in A_p(n)\) satisfies the subordination condition \(f'(z)/pz^{p-1}\prec[(1+z)/(1-z)]^\gamma\), where \(0<\gamma\leq 1\), \(z\in U\), then \[ \mathrm{Re}\left\{(1-\delta)\left(\frac{f'(z)}{pz^{p-1}}\right)^{\frac{1}{\gamma}}+\delta\left(1+\frac{zf''(z)}{f'(z)}\right)\right\}>\rho, \] where \(|z|<r_n(p,\gamma,\delta,\rho)\), \(0<\delta\leq 1, 0\leq\rho<1\) and \(r_n(p,\gamma,\delta,\rho)\) is the root in \((0,1)\) of the equation \[ [1+\rho-(p+1)\delta]r^{2n}-2(1-\delta+n\delta\gamma)r^n+1-\rho+(p-1)\delta=0. \]