Abstract
This article deals with some functional inequalities involving Struve function, generalized Struve function, and modified Struve functions. We aim to find the convexity of the integral operator defined by Struve function, generalized Struve function, and modified Struve functions.
1. Introduction
We denote by the class of functions which are analytic in the open unit disc and of the formLet denote the class of all functions in which are univalent in . Also let and be the subclasses of consisting of all functions which map onto a star shaped with respect to origin and convex domains, respectively. These classes are defined as Ozaki [1] showed that if a function is analytic in and of the form (1) such that and if either or then is univalent and convex in at least one direction in . It shows that the constants and are, in a certain sense, the best possible constants. In [2], for , consider the class consisting of locally univalent functions that satisfy the condition for That is, It can be seen that if and only if . Since whenever , it readily follows that the class is in , for , so in particular, the functions in are univalent functions. For some more detail about these conditions, see [3, 4].
Now we consider the second-order inhomogeneous differential equation The solution of the homogeneous part is Bessel functions of order , where is real or complex number. For some more details, see [5]. The particular solution of the inhomogeneous equation defined in (6) is called the Struve function of order . It is defined asNow we consider the differential equationEquation (8) differs from (6) in the coefficients of . Its particular solution is called the modified Struve functions of order and is given byAgain consider the second-order in-homogenous differential equationwhere Equation (10) generalizes (6) and (8). In particular for , we obtain (6) and for , we obtain (8). Its particular solution has the series form and is called the generalized Struve function of order . This series is convergent everywhere. We take the transformation where and , where denotes the Gamma function. The function is analytic in the whole complex plane and satisfies the differential equation The function unifies the Struve functions and modified Struve functions.
The function was introduced and studied by Orhan and Yagmur [6] and further investigated by [7–9]. In last few years, many mathematicians have set the univalence criteria of several those integral operators which preserve the class . By using a variety of different analytic techniques, operators, and special functions, several authors have studied univalence criterion. Recently Din et al. [10] studied the univalence of integral operators involving generalized Struve functions. These operators are defined as follows:In this paper, our main aim is to study the convexity and univalence of the integral operatorsApplications of Struve functions occur in water wave and surface wave problems, unsteady aerodynamics, resistive MHD instability theory, and optical diffraction. More recently, Struve functions appeared in particle quantum dynamic studies of spin decoherence and nanotubes electrodynamics, potential theory, and optics. For more details, we refer to see [11].
Lemma 1 (see [6]). If and , are so constrained that , then the function satisfies the following inequalities:(i),(ii)
2. Main Results
In this section, we find the convexity of these integral operators defined by generalized Struve functions, by using above lemma and the following inequalities. We also use Ozaki’s condition for the univalence of these operators.
2.1. Convexity Criteria for Integral Operators
Theorem 2. Let and be so constrained that , where . Consider the function , defined as Also suppose that and be positive real number and that satisfies the inequality then defined in (15) is in class and
Proof. It is clear that as it satisfies the condition Differentiating (15), we obtainClearly Differentiating logarithmically and after simple calculations, it follows thatUsing inequality (i) of Lemma 1, it is easy to see that so By putting the above inequality in (23), we have For and , we consider that the function defined by is decreasing , and therefore Hence which completes the proof that
Theorem 3. Let and be so constrained that where . Consider the function , defined as Also suppose that and be positive real number and that satisfies the inequality then defined in (16) is in class , where
Proof. Differentiating (16), we obtainClearly Differentiating logarithmically and after simple calculations, we obtainBy using inequality (ii) of Lemma 1, we have so Using above relation in (36), we have For and ,we consider that the function defined by is decreasing , and therefore Hence We have which completes the proof that
3. Some Special Cases of Struve Functions and Modified Struve Functions
3.1. Struve Functions
By setting in (11), we obtain the Struve function of first kind of order , denoted by defined in (7). Let be defined as We observe that
Corollary 4. Let , where Consider the function defined byAlso let and be positive real numbers. Suppose that these numbers satisfy the following inequality.then the function defined asis in class , where In particular
(i) First at , the functionis in class for , where (ii) For , then the function is in class for , where
(iii) For , the functionis in class for , where .
Corollary 5. Let where Consider the function defined by (46) and and be positive real numbers such that these numbers satisfy the inequalitythen the function defined asis in class , where and In particular, one has the following:
(i) For , thenis in class for , where
(ii) For , the functionis in class for , where .
(iii) For , the functionis in class for , where
3.2. Modified Struve Functions
We obtain the modified Struve function of first kind of order , denoted by , defined by (9), by putting in (11). Define a function by We observe that By making use of this particular value, we have the following assertions.
Corollary 6. Let where Consider the function defined by and and be positive real numbers. Suppose that these numbers satisfy the inequality then the function defined asis in class , where (i) In particular for , the integral operatoris in class for , where
Corollary 7. Let , where Consider the function defined by , and be positive real numbers. Suppose that these numbers satisfy the relation then the function defined as is , where In particular for , the functionis in class for , where
4. Locally Univalence Criteria
Now, we find the locally univalent criteria of the integral operators involving generalized Struve functions defined in (15), (16).
Theorem 8. Let and such that , where . Consider the function , defined as Also suppose that and be positive real number and that satisfies the inequality then defined in (16) is in class , where
Proof. Differentiating (16), we obtain Differentiating logarithmically, we obtainBy using (ii) of Lemma 1, it follows that Therefore, we haveFor and , we consider the function defined by is decreasing ; therefore This implies that Let Hence Here , which completes the proof.
Theorem 9. Let and be such that , where . Consider the function , defined as Also suppose that and are positive real number and that satisfies the inequality then defined in (15) is in class , where
Proof. Differentiating (15), we haveDifferentiating logarithmically, we obtainNow consider (i) of Lemma 1, we obtain Therefore For and , we consider the function defined by therefore This implies that Let Hence Here , which completes the proof.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors would like to acknowledge the heads of their institutions for their supportive role and research facilities. This research is conducted under the Project of Higher Education Commission of Pakistan (Project Ref: 5689/Punjab/NRPU/R&D/2016) and is partially supported by Sarhad University of Science and Information Technology, Peshawar, Pakistan.