Research Article

Moroccan Journal of Pure and Applied Analysis

, 2:8

First online:

Open Access

# Inclusion properties of Generalized Integral Transform using Duality Techniques

• Satwanti DeviAffiliated withDepartment of Applied Sciences, The NorthCap University Email author
• , A. SwaminathanAffiliated withDepartment of Mathematics, Indian Institute of Technology

## Abstract

Let $$W_\beta ^\delta (\alpha ,\gamma )$$ be the class of normalized analytic functions f defined in the region $$\left| z \right|<1$$ and satisfying

$$\text{Re}{{e}^{i\phi }}\left( (1-\alpha +2\gamma ){{(f/z)}^{\delta }}+\left( \alpha -3\gamma +\gamma \left[ (1-1/\delta (z{f}'/f)+1/\delta (1+z{f}''/{f}') \right] \right){{(f/z)}^{\delta }}(z{f}'/f)-\beta \right)>0,$$

with the conditions $$\alpha \ge 0,\beta {\rm{ }} \prec 1,\gamma \ge 0,\delta {\rm{ }} \succ 0$$ and ϕ ∈ ℝ. For a non-negative and realvalued integrable function λ(t) with $$\int_0^1 {\lambda (t)dt = {\rm{1}}}$$, the generalized non-linear integral transform is defined as

$$V_\lambda ^\delta (f)(z) = {\left( {\int_0^1 {\lambda (t){{(f(tz)/t)}^\delta }dt} } \right)^{1/\delta }}.$$

The main aim of the present work is to find conditions on the related parameters such that $$V_\lambda ^\delta (f)(z) \in W_{{\beta _1}}^{{\delta _1}}({\alpha _1},{\gamma _1}),$$ whenever $$f \in W_{{\beta _2}}^{{\delta _2}}({\alpha _2},{\gamma _2}).$$ Further, several interesting applications for specific choices of λ(t) are discussed.

### Key words and phrases

Integral Transforms Analytic functions Hypergeometric functions Convolution Duality techniques