Majorization problems for two subclasses of analytic functions connected with the Liu–Owa integral operator and exponential function

In the present paper, we investigate majorization properties for the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{\beta}^{\alpha}(p,\gamma)$\end{document}Mβα(p,γ) of uniformly starlike functions and the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N_{\beta}^{\alpha}(p,\theta)$\end{document}Nβα(p,θ) of spiral-like functions related to an exponential function, which are defined through the Liu–Owa integral operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q_{\beta,p}^{\alpha}$\end{document}Qβ,pα given by (1.5). Also, some special cases of our main results in a form of corollaries are shown.


Introduction and definitions
Let C be a complex plane and assume that A p denotes the class of analytic and p-valent functions of the form f (z) = z p + ∞ k=1 a k+p z k+p p ∈ N = {1, 2, . . .} (1.1) in the open unit disk U = z : z ∈ C and |z| < 1 .
In 1967, MacGregor [22] introduced the notion of majorization as follows.

Definition 1.1 Let f and g be analytic in U.
We say that f is majorized by g in U and write if there exists a function ϕ(z), analytic in U, satisfying ϕ(z) ≤ 1 and f (z) = ϕ(z)g(z) (z ∈ U).

Definition 1.2
For two analytic functions f and g in U, we say that f is quasi-subordinate to g in U and write if there exist two analytic functions ϕ(z) and ω(z) in U such that f (z) ϕ(z) is analytic in U and ϕ(z) ≤ 1, ω(0) = 0 and ω(z) ≤ |z| < 1 (z ∈ U), satisfying f (z) = ϕ(z)g ω(z) (z ∈ U). (1.3) and say that f is subordinate to g in U, denoted by (see [29]) In 1991, Ma and Minda [21] introduced the following function class S * (φ), which is defined by using the above subordination principle: where φ(z) is analytic and univalent in U and for which φ(U) is convex with φ(0) = 1 and (φ(z)) > 0 for z ∈ U. We notice that, for choosing a suitable function φ(z), the class S * (φ) reduces to one of the well-known classes of functions. For instance: (i) If we take then we obtain the class which was introduced by Janowski [16]. As a special case, for A = 1 -2α and B = -1, we have the class S * (1 -2α, -1) = S * (α) of starlike functions of order α (0 ≤ α < 1). Further, for A = 1 and B = -1, we have the familiar class S * (1, -1) = S * of starlike functions in U.
A majorization problem for the normalized class of starlike functions has been investigated by MacGregor [22] and Altintas et al. [1] (see also [2]). Recently, many researchers have studied several majorization problems for univalent and multivalent functions or meromorphic and multivalent meromorphic functions, which are all subordinate to certain function φ(z) = 1+Az 1+Bz (-1 ≤ B < A ≤ 1), involving various different operators; the interested reader can, for example, see [13-15, 18, 26, 27, 30, 31, 33]. However, we note that there is no article dealing with the above-mentioned problems for functions which are subordinate to φ(z) = e z . Hence, in the present paper, we investigate the problems of majorization of the classes M α β (p, γ ) and N α β (p, θ ) defined by the Liu-Owa integral operator Q α β,p given by (1.5), which are related to an exponential function.

Some corollaries
As a special case of Theorem 2.1, when p = 1, we get the following result.
Taking θ = 0 in Theorem 3.1, we state the following corollary.

Conclusions
In this paper, we investigate the problems of majorization of the classes M α β (p, γ ) and N α β (p, θ ) defined by the Liu-Owa integral operator Q α β,p given by (1.5), which are also related to an exponential function. The results obtained generalize and unify the theory of majorization in geometric function theory. In addition, we notice that, if we put p = 1 and α = 1, β = δ in Theorems 2.1 and 3.1, as well as Corollaries 4.2 and 4.3 of this paper, respectively, then we easily get the corresponding majorization results for the Jung-Kim-Srivastava integral operator Q α β and the generalized Libera operator J δ,p (δ > -p; p ∈ N), which are mentioned in the Introduction.