Multivalent Functions Related with an Integral Operator

In this present paper, we introduce and explore certain new classes of uniformly convex and starlike functions related to the Liu–Owa integral operator. We explore various properties and characteristics, such as coefficient estimates, rate of growth, distortion result, radii of close-to-convexity, starlikeness, convexity, and Hadamard product. It is important to mention that our results are a generalization of the number of existing results in the literature.


Introduction
Let C denote the complex plane and assume that A p denotes the class of p-valent function of the form which are analytic in the open unit disc ⋓ � ω: ω ∈ C { and |ω| < 1. Specially, for p � 1, we denote A�A 1 .

Theorem 1. A function λ(ω) given by (1) is in the class
where and Proof. It suffices to show that inequality (11) holds true. As we know, (18) en, inequality (11) may be written as which can be written as R(A(ω)/B(ω)) ≥ α, where and en, we have International Journal of Mathematics and Mathematical Sciences Now, Also, Using (23) and (24), then we can obtain the following inequality: e last expression is bounded below by 0 if where C t+p , D t+p , C p , and D p are given by (13)- (16), respectively.
Proof. In view of eorem 2, we need only to show that Since λ is a function of form (1) with the argument properties given in the class £ η and setting ω � re iη in the above inequality, we have Letting r ⟶ 1 − (29) leads to the desired inequality: is an external function for (27).
ese results are sharp for the extremal function λ(ω) given by (31).

Proof
(i) Given λ ∈ £ η and λ is starlike of order ϰ, we have For the left-hand side of (49), we have Use the fact that λ ∈ k − £U η (a, b, p, α, β, μ, ]) if and only if We can say (49) is true if or, equivalently, which is required.
(ii) Using the fact that λ is convex if and only if ωλ ′ (ω) is starlike, we can prove (ii) on similar lines to the proof of (i).

Conclusion
In our current investigation, we have presented and studied thoroughly some new subclasses of p− valent functions related with uniformly convex and starlike functions, in connection with the Liu-Owa integral operator Q a b,p λ(ω) given by (8). We have obtained sufficient and necessary conditions in relation to these classes, including growth, distortion theorem, and radius problem. Some special cases have been discussed as applications of our main results. e techniques and ideas of this paper may stimulate for further research in this area of knowledge.

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Authors' Contributions
SH came with the main thoughts and helped to draft the manuscript, SGAS and IA proved the main theorems, and SN and MD revised the paper. All authors read and approved the final manuscript.