Advanced Search

Let A denote the class of analytic functions with the normalization f(0) = f'(0) - 1 = 0 in the open unit disc U = {z : |z| < 1}. Set [...] and define ∞nλ, μ in terms of the Hadamard product [...] . In this paper, we introduce several subclasses of analytic functions defined by means of the operator Inλ, μ A → A, given by [...] . Inclusion properties of these classes and the classes involving the generalized Libera integral operator are also considered.

Let $A$ denote the class of analytic functions with the normalization $f\left(0\right)=f^{\text{'}}\left(0\right)-1=0$ in the open unit disc $U=\{z:\left|z\right|<1\}$.  Set $$f_{\lambda }^n\left(z\right)=z+\sum _{k=2}^{\infty }{[1+\lambda (k-1)]}^n{z}^k(n\in N_0;\lambda \ge 0;z\in U),$$ and define $f_{\lambda ,\mu }^n$ in terms of the Hadamard product $$f_{\lambda }^n\left(z\right)*f_{\lambda ,\mu }^n=\frac{z}{{(1-z)}^{\mu }}(\mu >0;z\in U).$$ In this paper, we introduce several subclasses of analytic functions defined by means of the operator $I_{\lambda ,\mu }^n:A⟶A$, given by $$I_{\lambda ,\mu }^{n}f\left(z\right)=f_{\lambda ,\mu }^n\left(z\right)*f\left(z\right)(f\in A;n\in N_{0;}\lambda \ge 0;\mu >0).$$ Inclusion properties of these classes and the classes involving the generalized Libera integral operator are also considered.

2000 Mathematics Subject Classification: 30C45