Duality and Integral Transform of a Class of Analytic Functions

Inclusion properties of Generalized Integral Transform using Duality Techniques

“…This operator is new in the literature whereas for the particular cases of this operator many interesting results are available. For example, when δ = 1, the operator was considered by several authors (see, for example [1,8,18]). Now, using this generalized operator the next result is given.…”

“…Since (4.5) is negative when 1 ≤ δ ≤ α, therefore it remains to prove that λ(t) ≥ 0, λ ′ (t) ≤ 0 and λ ′′ (t) ≥ 0, which is clearly true when −1 < a < b or −1 < b < a and this completes the proof. better bound for a as compared to [18,Theorem 5.7] but for γ = 0, the bound is weaker (see also [1,Theorem 5.3] for the case ξ = 0). 2.…”

“…Further study of this problem, where the operator (1.1) carries generalization of the functional class P β involving linear combinations of the functionals f (z)/z, f ′ (z) and zf "(z) to S * (ξ) were carried by many researchers in the recent future. 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]).…”

“…When δ = 1 and γ > 0, then Theorem 4.2 gives better range for b when compared to[18, Theorem 5.1] (see also[1, Theorem 5.5] for the case ξ = 0). 2.…”

Starlikeness and Convexity of Certain Integral Transforms by using Duality Technique

“…We obtain the bound for a as, −1 < a ≤ 0; a = b or a < b. This range for a is smaller than the range obtain in [15,Theorem 5.3] (see also [9,Theorem 3.2]). For the case when ξ = 1, Theorem 4.4 provides weaker bounds for a when compared to [16, Theorem 5.6].…”