Application of duality techniques to starlikeness of weighted integral transforms

Inclusion properties of Generalized Integral Transform using Duality Techniques

“…2] for the case γ = 0. This is because the condition given in (2.11) contains the function (1 + t)(1 − t) 1+2ξ whereas in [12], the result contains the function (log(1/t)) 1+2ξ in its denominator part. Both the functions are decreasing and tends to 0 as t → 1.…”

“…the integral operator (1.2) defined by the above weight function λ(t) is known as generalized Bernardi operator denoted by (B δ c ). This operator is the particular case of the generalized integral operators considered by [12] that follows in the sequel. For δ = 1, this operator was introduced by S. D. Bernardi [5].…”

“…Then the integral operator (1.2) defined by the above weight function λ(t) is the known as generalized Komatu operator denoted by (F δ k, p ). This integral operator was considered in the work of A. Ebadian [12]. When δ = 1, the operator is reduced to the one introduced by Y. Komatu [15].…”

“…then the integral operator (1.2) defined by the above weight function λ(t) is the known as generalized Hohlov operator denoted by H δ a,b,c . This integral operator was considered in the work of A. Ebadian [12]. When δ = 1, the reduced integral transform can be represented as the convolution of the normalized hypergeometric function with the analytic function z 2 F 1 (a, b; c; z) * f (z), which was introduced in the work of Y. C. Kim and F. Ronning [14] and studied by several authors later.…”

“…All these combinations were unified in a class introduced in [1] and for the most general result in this direction, see [8,16,18]. Since such combinations and the generalization of the operator encompass large number of previous results such as univalency, subordination and positivity results on classes of functions, we are interested in considering the following generalized operator The integral operator defined in (1.2) and its more generalized form was considered in the work of I. E. Bazilevič [4] (see also [12,19]). Note that when δ = 1, the operator (1.2) reduces to (1.1).…”