On Fekete-Szegő problems for certain subclass of analytic functions (Q2879670)

Let \(A\) denote the class of all functions \(f\) of the form \(f(z) = z + a_2z^2 + a_3z^3 + \dots\), analytic in the open unit disk \(U\). For \(m,n \in \mathbb{N} \cup \{0\}\) and \(\lambda_2 \geq \lambda_1 \geq 0\) the authors define the following differential operator NEWLINE\[NEWLINED_{\lambda_1,\lambda_2}^{n,m}\, f(z) = z + \sum_{k=n+1}^{\infty}\,\left[\frac{1+(\lambda_1+\lambda_2) (k-1)}{1+\lambda_2(k-1)}\right]^m\,C(n,k)a_kz^k,NEWLINE\]NEWLINE where \(z \in U\) and \(f \in A\). Next they consider the class \(M_{\lambda_1,\lambda_2}^{n,m}\) consisting of all functions \(f \in A\) for which NEWLINE\[NEWLINE\frac{z(D_{\lambda_1,\lambda_2}^{n,m}\, f(z))'}{D_{\lambda_1,\lambda_2}^{n,m}\, f(z)}NEWLINE\]NEWLINE is subordinate to the function \(\phi(z)\) in \(U\), where \(\phi\) is a fixed analytic function with positive real part in \(U\), \(\phi(0)=1\), \(\phi'(0) > 0\) and \(\phi(U)\) is a domain symmetric with respect to the real axis and starlike with respect to the point \(1\).NEWLINENEWLINEThe authors obtain Fekete-Szegő inequalities for functions in the class \(M_{\lambda_1,\lambda_2}^{n,m}\) and in some related classes defined by convolution.