Differential subordinations concerning starlike functions
Let \(A\) denote the family of functions \(f\) that are analytic in the disk \(| z|<1\) and normalized by \(f(0)= f'(0)- 1= 0\). Let \(S^*(\rho)\), \(0\leq \rho< 1\) be the class of those functions from \(A\) that are univalent and starlike of order \(\rho\) in \(| z|< 1\). The differential subordination technique is used to prove the sufficient conditions for the function \(f\) from \(A\) to be in \(S^*(\rho)\). Theorems: 1. Suppose that \(\alpha'\), \({1\over 2}< \alpha'< 1\), is the unique root of the equation: \[ \mu= (1- \alpha)\text{cot}\left[(2\alpha- 1){\pi\over 2}\right], \] for a suitable fixed \(\mu>0\). If \(f\in A\) and satisfy \[ |\arg f'(z)(f(z)/z)^{\mu-1}|< \alpha'{\pi\over 2} \] then \(f\in S^*(0)\). 2. Suppose that \(\mu> 0\), while \(\lambda\) is complex, \(|\lambda|\leq (\mu+ 1)[2\mu(\mu+1)+ 1]^{-{1\over 2}}\). If \(f\in A\) and \(f'(z)(f(z)/z)^{\mu-1}\) is subordinate to \(1+ \lambda z\) in \(| z|<1\), then \(f\in S^*(\beta)\), where \(\beta= [\mu+1- | \lambda|(2\mu(\mu+ 1)+1)^{{1\over 2}}]/(\mu+1+ | \lambda|)\). 3. Let \(\alpha\) be a complex number of positive real part and \(|\alpha+1| \geq \sqrt 5{\gamma\over 2}\). If \(f\in A\) and \(| f'(z)+ \alpha zf''(z)- 1|< \gamma\) in \(| z|< 1\), then \(f\in S^*(\rho)\), where \(\rho= {2| \alpha+1|- \sqrt 5\gamma\over 2|\alpha+1|+\gamma}\).
scientific article; zbMATH DE number 618133
scientific article