Analytic Functions of Complex Order Defined by New Differential Operator
Analytic Functions of Complex Order Defined by New Differential Operator is a scholarly work, published in 2018 in ''Open Journal of Mathematical Sciences''. The main subjects of the publication include stock order, pure mathematics, mathematical analysis, differential, extreme point, Differential inclusion, relation, analytic function, distortion, operon, differential operator, geometric function theory, applied mathematics, and mathematics. The authors introduce and study the classes \\(S_{n,\\mu}(\\gamma,\\alpha,\\beta,\\) \\(\\lambda,\\nu,\\varrho,\\mho)\\) and \\(R_{n,\\mu}(\\gamma,\\alpha,\\beta,\\lambda,\\nu,\\varrho,\\mho)\\) of functions \\(f\\in A(n)\\) with \\((\\mu)z(D^{\\mho+2}_{\\lambda,\\nu,\\varrho}(\\alpha,\\omega)f(z))^{'} \\) \\(+(1-\\mu)z(D^{\\mho+1}_{\\lambda,\\nu,\\varrho}(\\alpha,\\omega)f(z))^{'}\\neq0\\), where \\(\\nu>0,\\varrho,\\omega,\\lambda,\\alpha,\\mu \\geq0, \\mho\\in N_{0}, z\\in U\\) and \\(D^{\\mho}_{\\lambda,\\nu,\\varrho}(\\alpha,\\omega)f(z):A(n)\\longrightarrow A(n),\\) is the linear differential operator, newly defined as \n\\( D^{\\mho}_{\\lambda,\\nu,\\varrho}(\\alpha,\\omega)f(z)=z-\\sum_{k=n}^{\\infty}\\left( \\dfrac{\\nu+k(\\varrho+\\lambda)\\omega^{\\alpha}}{\\nu} \\right)^{\\mho} a_{k+1}z^{k+1}. \\) \nSeveral properties such as coefficient estimates, growth and distortion theorems, extreme points, integral means inequalities and inclusion relation for the functions included in the classes \\(S_{n,\\mu} (\\gamma,\\alpha,\\beta,\\lambda,\\nu,\\varrho,\\mho,\\omega)\\) and \\(R_{n,\\mu}(\\gamma,\\alpha,\\beta,\\lambda,\\nu,\\varrho,\\mho,\\omega)\\) are given.