Abstract
Using the third-order differential subordination basic results, we introduce certain classes of admissible functions and investigate some applications of third-order differential subordination for p-valent functions associated with generalized fractional differintegral operator.
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1 Introduction and preliminaries
Let \(\mathcal{H}(\mathbb{U})\) denote the space of analytic functions in the unit disk \(\mathbb{U}:= \{z\in \mathbb{C}: \vert z \vert <1 \}\), \(\mathcal{H}[a,n]\) denote the subclass of functions \(f\in \mathcal{H}( \mathbb{U})\) of the form
and \(\mathcal{H}_{p}:=\mathcal{H}[0,p]\). Also, let \(\mathcal{A}(p)\) be the subclass of functions \(f\in \mathcal{H}(\mathbb{U})\) of the form
and set \(\mathcal{A}:=\mathcal{A}(1)\).
For two functions \(f, g\in \mathcal{H}(\mathbb{U})\), we say that the function f is subordinate to g, written \(f\prec g\), if there exists a Schwarz function w, analytic in \(\mathbb{U}\) with \(w(0)=0\) and \(\vert w(z) \vert <1\) such that \(f(z)=g(w(z))\) for all \(z\in \mathbb{U}\). Furthermore, if the function g is univalent in \(\mathbb{U}\), then we have the following equivalence (see [9] and [12]):
Let \(\phi (r,s,t;z):\mathbb{C}^{4}\times \mathbb{U}\rightarrow \mathbb{C}\) and h be univalent in \(\mathbb{U}\). If p is analytic in \(\mathbb{U}\) and satisfies the third-order differential subordination
then p is called a solution of the differential subordination (1.2). The univalent function q is said to be a dominant of (1.2) if \(p(z)\prec q(z)\) for all p that satisfy (1.2). A dominant q̃ is called the best dominant if \(\widetilde{q}(z)\prec q(z)\) for all dominants q of (1.2).
We recall here the following generalized fractional integral and generalized fractional derivative operators due to Srivastava et al. [22] (see also [16, 17]).
Definition 1.1
([22, Definition 3])
For \(\lambda >0\) and μ, η real numbers, the Srivastava–Saigo–Owa hypergeometric fractional integral operator \(I_{0,z}^{\lambda ,\mu , \eta }\) is defined by
where f is an analytic function in a simply-connected region of the complex z-plane containing the origin with the order \(f(z)=O ( \vert z \vert ^{ \varepsilon } )\), \(z\to 0\), where \(\varepsilon >\max \{0,\mu -\eta \}-1\) and the multiplicity of \((z-\zeta )^{\lambda -1}\) is removed by requiring \(\log (z-\zeta )\) to be real when \(z-\zeta >0\). Also, Γ is the well-known gamma function, while the function F denotes the Gauss hypergeometric function, that is,
and its analytic continuation into \(\vert \arg (1-z) \vert <\pi \) and \((a)_{n}=\varGamma (a+n)/\varGamma (a)\).
Definition 1.2
Under the hypotheses of Definition 1.1, the Srivastava–Saigo–Owa hypergeometric fractional derivative operator \(J_{0,z}^{\lambda ,\mu ,\eta }\) is defined by
where the multiplicity of \((z-\zeta )^{-\lambda }\) is removed as in Definition 1.1.
We note that
where \(D_{z}^{-\lambda }\) denotes the fractional integral operator, and \(D_{z}^{\lambda }\) denotes the fractional derivative operator studied by Owa [13].
In relation to the Srivastava–Saigo–Owa hypergeometric fractional derivative operator, Goyal and Prajapat [10] (see also [15]) defined the operator
Thus, for a function \(f\in \mathcal{A}(p)\) of the form (1.1), we have
where \(_{q}F_{s}\), with \(q\leq s+1\) and \(q,s\in \mathbb{N}_{0}:= \mathbb{N} \cup \{0\}\), is the well-known generalized hypergeometric function (for more details, see [14] and [20]) and \((\nu )_{n}\) is the Pochhammer symbol defined by
Let
and define the new function \([G_{p,\eta ,\mu }^{\lambda } ] ^{-1}\) by means of the Hadamard (or convolution) product
Using the above defined function, Tang et al. [24] (see also Aouf et al. [4, 7] and [8]) defined the operator \(H_{p, \eta ,\mu }^{\lambda ,\delta }:\mathcal{A}(p)\rightarrow \mathcal{A}(p)\) by
It is easy to check that, for a function \(f\in \mathcal{A}(p)\) of the form (1.1), we have
For \(k\in \mathbb{N}_{0}\) and \(\zeta >0\), Aouf et al. [6] defined the operator \(\mathcal{N}_{p,\lambda ,\mu ,\eta }^{m,\delta ,\zeta }: \mathcal{A} (p)\rightarrow \mathcal{A}(p)\) as follows:
and, in general,
Remark 1.1
1. We note that the operator \(\mathcal{N}_{p,\lambda ,\mu ,\eta }^{m, \delta ,\zeta }\) generalizes many other remarkable previously studied operators, like:
2. Also, we remark the following special cases of this operator:
Moreover, it is easy to verify from (1.3) that
and
To obtain our results, we need to use the following definitions and theorems.
Definition 1.3
([2, p. 441])
Let \(\mathcal{Q}\) be the set of all functions q that are analytic and univalent on \(\overline{\mathbb{U}}\setminus E(q)\), where
and are such that
Further, let the subclass of \(\mathcal{Q}\) for which \(q(0)=a\) be denoted by \(\mathcal{Q}(a)\) and \(\mathcal{Q}(0)=:\mathcal{Q}_{0}\).
Definition 1.4
([2, p. 449])
Let Ω be a subset of \(\mathbb{C}\), \(q\in \mathcal{Q}\) and \(n\geq 2\). The class of admissible operators \(\varPsi _{n}[\varOmega ,q]\) consists of those functions \(\psi :\mathbb{C} ^{4} \times \mathbb{U}\rightarrow \mathbb{C}\) that satisfy the following admissibility condition
whenever
and
Lemma 1.1
([2, p. 449])
Let Ω be a subset of \(\mathbb{C}\), \(\psi \in \varPsi _{n}[\varOmega ,q]\) and \(p\in \mathcal{H}[a,n]\) with \(n\geq 2\). If \(q\in \mathcal{Q}(a)\) and satisfies the following conditions
then
implies \(p(z)\prec q(z)\).
The aim of the present article is to use the recent works by Tang et al. (see [25, 26]) to systematically investigate the third-order differential subordination general theory to a suitable classes of admissible functions. We obtained new results for a wide class of operators defined by convolution products with Srivastava–Saigo–Owa generalized fractional integral and generalized fractional derivative operators. Our results give interesting new properties and, together with other papers that appeared in the last years, could emphasize the perspective of the importance of the third-order subordination theory and the Srivastava–Saigo–Owa generalized operators.
We emphasize that in recent years, several authors obtained many interesting results involving different linear and convolution operators associated with second-order differential subordinations (see [19, 23]) and regarding the third-order differential subordinations (see [21]) for the above mentioned operator.
2 Main results
Unless otherwise mentioned, we assume throughout this paper that \(f\in \mathcal{A}(p)\), \(\zeta >0\), \(p\geq 2\), \(w\in \partial \mathbb{U}\setminus E(q)\), \(\theta \in [0,2\pi ]\) and \(z\in \mathbb{U}\).
Definition 2.1
Let Ω be a subset of \(\mathbb{C}\) and \(q\in \mathcal{Q} _{0}\). The class of admissible operators \(\varPhi _{p}[\varOmega ,q]\) consists of those functions \(\phi :\mathbb{C}^{4}\times \mathbb{U} \rightarrow \mathbb{C}\) that satisfy the following admissibility condition:
whenever
and
Theorem 2.1
Let Ω be a subset of \(\mathbb{C}\) and \(\phi \in \varPhi _{p}[ \varOmega ,q]\). If \(q\in \mathcal{Q}_{0}\) satisfies the following conditions:
then
implies
Proof
Defining the function p by
then p is analytic in \(\mathbb{U}\). Differentiating three times (2.3) with respect to z and using (1.4), we obtain the following relations, respectively:
Letting
we will define the transformation \(\psi :\mathbb{C}^{4}\times \mathbb{U} \rightarrow \mathbb{C}\) by
Then, using relations (2.3), (2.4), (2.5) and (2.6), we have
Since
and
the admissibility condition for \(\phi \in \varPhi _{p} [\varOmega ,q ]\) of Definition 2.1 is equivalent to the admissibility condition for ψ as given in Definition 1.4. Thus, the proof follows from Lemma 1.1 by setting \(a=0\), \(n=p\) and \(a_{p}=1\). □
The next result is an extension of Theorem 2.1 for the case when the behavior of q on \(\partial \mathbb{U}\) is unknown.
Corollary 2.1
Let Ω be a subset of \(\mathbb{C}\) and q be univalent in \(\mathbb{U}\) with \(q\in \mathcal{Q}_{0}\). Let \(\phi \in \varPhi _{p} [ \varOmega ,q_{\rho } ]\) for some \(\rho \in (0,1 )\), where \(q_{\rho }(z)=q(\rho z)\). If \(q_{\rho }\) satisfies the following conditions:
then
implies
Proof
From Theorem 2.1 we obtain \(\mathcal{N}_{p,\lambda ,\mu , \eta } ^{k,\delta ,\zeta }f(z)\prec q_{\rho }(z)\) and since \(q_{\rho }(z)\prec q(z)\), we conclude that \(\mathcal{N}_{p,\lambda , \mu ,\eta }^{k,\delta ,\zeta }f(z)\prec q(z)\). □
If \(\varOmega \neq \mathbb{C}\) is a simply connected domain, then \(\varOmega =h( \mathbb{U})\) for some conformal mapping h of \(\mathbb{U}\) onto Ω. In this case, the class \(\varPhi _{p} [h( \mathbb{U}),q ]\) will be written as \(\varPhi _{p} [h,q ]\). The following two results are direct consequences of Theorem 2.1 and Corollary 2.1.
Corollary 2.2
Let \(\phi \in \varPhi _{p}[h,q]\), where h is univalent in \(\mathbb{ U}\) and suppose that \(q\in \mathcal{Q}_{0}\) satisfies conditions (2.1). Then
implies
Corollary 2.3
Let q be univalent in \(\mathbb{U}\) with \(q\in \mathcal{Q}_{0}\) and \(\phi \in \varPhi _{p} [h,q_{\rho } ]\) for some \(\rho \in ( 0,1 )\), where \(q_{\rho }(z)=q(\rho z)\). If \(q_{\rho }\) satisfies conditions (2.9), then the subordination (2.10) implies that
We next show the relation between the best dominant of a differential subordination and the solution of a corresponding differential equation.
Corollary 2.4
Let h be univalent in \(\mathbb{U}\) and ψ be given by (2.8) where \(\phi \in \varPhi _{p}[h,q]\). Suppose that the differential equation
has a solution q with \(q\in \mathcal{Q}_{0}\) that satisfies conditions (2.1). Then subordination (2.10) implies that
and q is the best dominant of (2.10).
Proof
Since
then p is a solution of (2.11), and from Corollary 2.2 we obtain that \(p(z)\prec q(z)\), that is, q is a dominant of (2.11). Also,
which means that q is the best dominant of (2.11). □
3 Special Cases
We specialize the class of admissible functions and corresponding theorems for the case when \(q(\mathbb{U})\) is the disk \(\mathbb{U} _{M}:= \{ w\in \mathbb{C}: \vert w \vert < M \}\). First, we remark that the function
is univalent in \(\overline{\mathbb{U}}\) and satisfies \(q(\mathbb{U})= \mathbb{U}_{M}\), \(q\in \mathcal{Q}_{0}\) and \(E(q)=\emptyset \).
Definition 3.1
Let Ω be a subset of \(\mathbb{C}\) and q be given by (3.1). The class of admissible operators \(\varPhi _{p} [ \varOmega ,M ]\) consists of those functions \(\phi : \mathbb{C}^{4} \times \mathbb{U}\rightarrow \mathbb{C}\) that satisfy the following admissibility condition:
whenever
and
where \(\operatorname{Re} (Le^{-i\theta } )\geq p(p-1)M\) and \(\operatorname{Re} (Ne^{-i\theta } )\geq 0\) for all \(\theta \in [0,2\pi ]\) and \(p\geq 2\).
Using this definition of the class of admissible functions, from Theorem 2.1 we obtain the following result.
Corollary 3.1
Let Ω be a subset of \(\mathbb{C}\) and \(\phi \in \varPhi _{p} [\varOmega ,M ]\). If we suppose that
and the function q is given by (3.1), then
implies
For the special case when \(\varOmega =q(\mathbb{U})= \{w\in \mathbb{C} : \vert w \vert < M \}\), Corollary 3.1 reduces to the following corollary.
Corollary 3.2
Let \(\phi \in \varPhi _{p} [q(\mathbb{U}),M ]\) and suppose that the function q given by (3.1) satisfies condition (3.3). Then
implies
Let \(\phi (\alpha _{1},\beta _{1},\gamma _{1},\varepsilon _{1};z ) = \alpha _{1}+\beta _{1}\) and \(\varOmega =h(\mathbb{U})\), where \(h(z)=2Mz\). We will show that \(\phi \in \varPhi _{p} [h(\mathbb{U}),M ]\) by proving that condition (3.2) is satisfied. Thus,
where \(\operatorname{Re} (Le^{-i\theta } )\geq p(p-1)M\), \(\operatorname{Re} (Ne^{-i\theta } )\geq 0\) for all \(\theta \in [0,2\pi ]\) and \(p\geq 2\).
Suppose that A and B are two complex-valued functions defined on \(\mathbb{U}\) that satisfy \(\operatorname{Re}A(z)>0\), \(\operatorname{Re}B(z)>0\) for all \(z\in \mathbb{U}\). Let \(\phi (\alpha _{1},\beta _{1},\gamma _{1},\varepsilon _{1};z )=1+A(z)\alpha _{1}+B(z) \beta _{1}\) and \(\varOmega =h(\mathbb{U})\), where \(h(z)=z\). We will show that \(\phi \in \varPhi _{p} [h(\mathbb{U}),M ]\) by proving that condition (3.2) is satisfied. Since
where \(\operatorname{Re} (Le^{-i\theta } )\geq p(p-1)M\), \(\operatorname{Re} (Ne^{-i\theta } )\geq 0\) for all \(\theta \in [0,2\pi ]\) and \(p\geq 2\).
Let \(A:\mathbb{U}\rightarrow (1,+\infty )\), \(B:\mathbb{U}\rightarrow (0,+\infty )\), \(\phi (\alpha _{1},\beta _{1},\gamma _{1}, \varepsilon _{1};z ) =\alpha _{1}+\beta _{1}+A(z)\gamma _{1}+B(z) \varepsilon _{1}\) and \(\varOmega =h(\mathbb{U})\), where \(h(z)=4Mz\). We will show that \(\phi \in \varPhi _{p} [ h(\mathbb{U}),M ]\) by proving that condition (3.2) is satisfied. Thus,
where \(\operatorname{Re} (Le^{-i\theta } )\geq p(p-1)M\), \(\operatorname{Re} (Ne^{-i\theta } )\geq 0\) for all \(\theta \in [0,2\pi ]\) and \(p\geq 2\), from Corollary 3.2 we have the following special case.
Example 3.1
If \(A:\mathbb{U}\rightarrow (1,+\infty )\), \(B:\mathbb{U} \rightarrow (0,+\infty )\) and \(f\in \mathcal{A}(p)\) such that
then
implies that
Let \(A,B:\mathbb{U}\rightarrow \mathbb{C}\), with \(\operatorname{Re} [ A(z)+B(z) ]>0\) for all \(z\in \mathbb{U}\), let \(C:\mathbb{U} \rightarrow (1,+\infty )\), \(D:\mathbb{U}\rightarrow (0,+ \infty )\), \(\phi ( \alpha _{1},\beta _{1},\gamma _{1},\varepsilon _{1};z )= A(z)\alpha _{1}+B(z)\beta _{1}+C(z)\gamma _{1}+D(z) \varepsilon _{1}\) and \(\varOmega =h( \mathbb{U})\), where \(h(z)=2Mz\). We will show that \(\phi \in \varPhi _{p} [ h(\mathbb{U}),M ]\) by proving that condition (3.2) is satisfied. Thus,
where \(\operatorname{Re} (Le^{-i\theta } )\geq p(p-1)M\), \(\operatorname{Re} (Ne^{-i\theta } )\geq 0\) for all \(\theta \in [0,2\pi ]\) and \(p\geq 2\), from Corollary 3.2 we have the following special case.
Example 3.2
Let \(A,B:\mathbb{U}\rightarrow \mathbb{C}\), with \(\operatorname{Re} [A(z)+B(z) ]>0\) for all \(z\in \mathbb{U}\), and \(C: \mathbb{U}\rightarrow (1,+\infty )\), \(D:\mathbb{U}\rightarrow (0,+ \infty )\). If \(f\in \mathcal{A}(p)\) such that
then
implies that
Let \(\phi (\alpha _{1},\beta _{1},\gamma _{1},\varepsilon _{1};z )=1+ \frac{ \gamma _{1}}{\alpha _{1}}+\frac{\varepsilon _{1}}{\alpha _{1}}\) and \(\varOmega = \{w\in \mathbb{C}:\operatorname{Re}w<3 \}\). We will show that \(\phi \in \varPhi _{p} [\varOmega ,M ]\) by proving that condition (3.2) is satisfied. Thus,
where \(\operatorname{Re} (Le^{-i\theta } )\geq p(p-1)M\) and \(\operatorname{Re} (Ne^{-i\theta } )\geq 0\) for all \(\theta \in [0,2\pi ]\) and \(p\geq 2\), from Corollary 3.2 we obtain the following.
Example 3.3
If \(f\in \mathcal{A}(p)\) such that
then
implies that
Remark 3.1
For different choices of k, δ, ζ, p, λ, μ, and η, we will obtain new results for different operators defined in the introduction.
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Acknowledgements
The authors are grateful to the reviewers of this article that gave valuable remarks, comments, and advice in order to revise and improve the results of the paper. Further, the first author would like to thank ITAM for the kind hospitality during her stay in Mexico.
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Zayed, H.M., Bulboacă, T. Applications of differential subordinations involving a generalized fractional differintegral operator. J Inequal Appl 2019, 242 (2019). https://doi.org/10.1186/s13660-019-2198-0
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DOI: https://doi.org/10.1186/s13660-019-2198-0