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Some subclasses of analytic functions of complex order defined by new differential operator

DOI: 10.5556/j.tkjm.43.2012.223-242

Keywords: Neighborhoods properties , Analytic functions , Inclusion properties , Identity function , $(n , delta)$-neighborhoods , Differential operator , convex function , coefficient estimates , growth and distortion theorems , Hadamard Product , Extreme Points , Integral M

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Abstract:

Let hskip 2pt $mathcal{A}(n)$ hskip 2pt denote hskip 2pt the hskip 2pt class hskip 2pt of hskip 2pt analytic hskip 2pt functions hskip 2pt $f$ hskip 2pt in hskip 2pt the hskip 2pt open hskip 2pt unit hskip 2pt disk hskip 2pt $U={z:|z|<1}$ hskip 2pt normalized hskip 2pt by hskip 2pt $f(0)=f'(0)-1=0.$ hskip 2pt In hskip 2pt this hskip 2pt paper, hskip 2pt we hskip 2pt introduce hskip 2pt and hskip 2pt study hskip 2pt the hskip 2pt classes hskip 2pt $S_{n, mu}(gamma, alpha, eta, lambda, mho)$ hskip 2pt and hskip 2pt $R_{n, mu}(gamma, alpha, eta, lambda, mho)$ hskip 2pt of hskip 2pt functions hskip 2pt $finmathcal{A}(n)$ with $(mu)z(D^{mho+2}_{lambda}(alpha, omega)f(z))'+(1-mu)z(D^{mho+1}_{lambda}(alpha, omega)f(z))' eq0$ and satisfy some conditions available in literature, where $finmathcal{A}(n), alpha, omega, lambda, mu geq0, mhoin mathbb{N}cup{0},,,zin U,$ and $D^{m}_{lambda}(alpha, omega)f(z): mathcal{A} ightarrow mathcal{A},$ is the linear fractional differential operator, newly defined as follows $$D^{m}_{lambda}(alpha, omega)f(z) = z+ sumlimits_{k=2}^{infty}a_{k}(1+(k-1)lambda omega^{alpha})^{m}z^{k}cdot$$ Several properties such as coefficient estimates, growth and distortion theorems, extreme points, integral means inequalities and inclusion for the functions included in the classes $S_{n, mu}(gamma, alpha, eta, lambda, mho, omega)$ and $R_{n, mu}(gamma, alpha, eta, lambda, mho, omega)$ are given.

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