In this article, we are mainly interested to find some covexity properties for certain families of integral operators involving Dini functions in the open unit disc. The main tool in the proofs of our results are some functional inequalities of Dini functions. Some particular cases involving Dini functions are also a part of our investigations

Mohsan Raza

Muhey U Din

Saddaf Noreen

International Journal of Analysis and Applications

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Convexity of Integral Operators Involving Dini Functions

International Journal of Analysis and ApplicationsVolume 16, Number 4 (2018), 462-471URL: https://doi.org/10.28924/2291-8639DOI: 10.28924/2291-8639-16-2018-462CONVEXITY OF INTEGRAL OPERATORS INVOLVING DINI FUNCTIONSSADDAF NOREEN, MUHEY U DIN AND MOHSAN RAZA∗Department of Mathematics, Government College University Faisalabad, Pakistan∗Corresponding author: mohsan976@yahoo.comAbstract. In this article, we are mainly interested to find some covexity properties for certain families ofintegral operators involving Dini functions in the open unit disc. The main tool in the proofs of our resultsare some functional inequalities of Dini functions. Some particular cases involving Dini functions are also apart of our investigations.1. INTRODUCTIONLet A denote the class of functions f of the formf(z) = z +∞∑n=2anzn, (1.1)which are analytic in the open unit disc U = {z : |z| < 1} and S denote the class of all functions whichare univalent in U . Let S∗ (α) , C (α) and K (α) denote the classes of starlike, convex and close-to-convexfunctions of order α and are defined as:S∗ (α) ={f : f ∈ A and Re(zf ′ (z)f (z))> α, z ∈ U , α ∈ [0, 1)},C (α) ={f : f ∈ A and Re(1 +zf ′′ (z)f ′ (z))> α, z ∈ U , α ∈ [0, 1)}andK (α) ={f : f ∈ A and Re(zf ′ (z)g (z))> α, z ∈ U , α ∈ [0, 1) , g ∈ S∗}.Received 2018-01-31; accepted 2018-04-06; published 2018-07-02.2010 Mathematics Subject Classification. 30C45, 33C10, 30C20, 30C75.Key words and phrases. analytic functions; convex functions; integral operators; Bessel functions; Dini functions.c©2018 Authors retain the copyrightsof their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.462Int. J. Anal. Appl. 16 (4) (2018) 463It is clear thatS∗ (0) = S∗, C (0) = C and K (0) = K.Special functions have great importance in pure and applied mathematics. The widely use of these functionshave attracted many researchers to work on the different directions. Geometric properties of special func-tions such as Hypergeometric functions, Bessel functions, Struve functions, Mittage-Lefller functions, Wrightfunctions and some other related functions is an ongoing part of research in geometric function theory. Werefer for some geometric properties of these functions [1, 2, 5, 6, 11] and references therein.The Bessel function of the first kind Jv is defined byJv(z) =∞∑n=1(−1)nn!Γ(v + n+ 1)(z2)2n+v, (1.2)where Γ stands for Euler gamma function. It is a particular solution of the second order linear homogeneousdifferential equationz2w′′(z) + zw′(z) + (z2 − v2)w(z) = 0,where v ∈ C. For some details see [2, 13]. Bessel functions are indispensable in many branches of pure andapplied mathematics. Thus, it is important to study their properties in many aspects. Recently Bariczet al [3] studied the close-to-convexity of Dini functions and some monotonicity properties and functionalinequalities for the modified Dini function are discussed in [4]. Further some geometric properties of Dinifunctions are studied in [3, 4, 7]. Now, we consider the normalized Dini functions qυ : U → C defined asqυ (z) = 2υΓ (υ + 1) z1−υ2((1− υ) Jυ(√z)+√zJ ′υ(√z))= z +∞∑n=1(−1)n (2n+ 1) Γ (υ + 1)4nn!Γ (υ + n+ 1)zn+1, z ∈ U . (1.3)The Pochhammer (or Appell) symbol, defined in terms of Euler’s gamma functions is given as (x)n =Γ(x+ n)/Γ(x) = x(x+ 1)...(x+ n− 1).Recently, Deniz et al. [8], Din et al. [9], Din et al. [10] and Srivastava et al. [12] have obtained sufficientconditions for the univalence of certain families of integral operators defined by Bessel, Dini, Struve andMittage-Leffler functions respectively. The families of integral operators are defined below:Fα1,...,αn,ζ (z) =ζz∫0tζ−1n∏i=1(fi(t)t) 1αidt1/ζ, (1.4)Gξ,n (z) =(nξ + 1)z∫0n∏i=1(fi (t))ξdt1/(nξ+1), (1.5)Hδ1,...,δn,µ (z) =µz∫0tµ−1n∏i=1(f ′i(t))δi dt1/µ(1.6)Int. J. Anal. Appl. 16 (4) (2018) 464andQλ (z) =λz∫0tλ−1(ef(t))λdt1/λ. (1.7)In this paper, we are mainly interested in the convexity of the integral operators involving Dini functionqv. These integral operators are defined asFv1,...,vn,α1,...,αn (z) =z∫0n∏i=1(qvi(t)t)αidt (1.8)andHv1,...,vn,δ1,...,δn (z) =z∫0n∏i=1(q′vi(t))δidt. (1.9)Now we prove some functional inequalities which are useful in establishing our main results.Lemma 1.1. Let v ∈ R and consider the normalized Dini function qυ : U → C, defined byqυ (z) = 2υΓ (υ + 1) z1−υ2((1− υ) Jυ(√z)+√zJ ′υ(√z)),where Jυ is the Bessel function of first kind. Then the following inequalities hold for all z ∈ U .(i)∣∣∣ zq′υ(z)qυ(z) − 1∣∣∣ ≤ 3v+64v2+5v−2 , v > −5+√578 ,(ii)∣∣∣ zq′′υ (z)q′υ(z) ∣∣∣ ≤ 3v+62v2+v−4 , v > −1+√334 .Proof. (i) By using the well known triangle inequality with the equalityΓ (v + 1)Γ (v + n+ 1)=1(v + 1) (v + 2) · · · (v + n)=1(v + 1)n, n ∈ Nand the inequality4nn! (v + 2)n−1 ≥43n (2n+ 1) (v + 2)n−1, n ∈ N,we obtain ∣∣∣∣q′υ (z)− qυ (z)z∣∣∣∣ ≤ 34 (v + 1)∞∑n=1(1v + 2)n−1=3 (v + 2)4 (v + 1)2 .Furthermore, if we use the reverse triangle inequality and the inequality4nn! (v + 2)n−1 ≥13(2n+ 1) (v + 2)n−1, n ∈ N,Int. J. Anal. Appl. 16 (4) (2018) 465then we get ∣∣∣∣qυ (z)z∣∣∣∣ =∣∣∣∣∣1 +∞∑n=1(−1)n (2n+ 1) Γ (υ + 1)4nn!Γ (υ + n+ 1)∣∣∣∣∣≥ 1− 34 (v + 1)∞∑n=0(1v + 2)n−1=4 (v + 1)2 − 3 (v + 2)4 (v + 1)2 .By combining the above inequalities, we get∣∣∣∣zq′υ (z)qυ (z) − 1∣∣∣∣ ≤ 3v + 64v2 + 5v − 2 , v > −5 +√578.(ii) By using the well known triangle inequality with equalityΓ (v + 1)Γ (v + n+ 1)=1(v + 1) (v + 2) · · · (v + n)=1(v + 1)n, n ∈ Nand the inequality4nn! (v + 2)n−1 ≥43(2n+ 1) (v + 2)n−1, n ∈ N,we have|zq′′υ (z)| ≤∣∣∣∣∣∞∑n=1n (n+ 1) (2n+ 1) Γ (υ + 1)4nn!Γ (υ + n+ 1)∣∣∣∣∣≤ 32 (v + 1)∞∑n=1(1v + 2)n−1=3 (v + 2)2 (v + 1)2 .Furthermore, if we use the reverse triangle inequality and the inequality4nn! (v + 2)n−1 ≥23(2n2 + 3n+ 1)(v + 2)n−1, n ∈ N,then we get|q′υ (z)|∣∣∣∣∣≥ 1−∞∑n=1(2n+ 1) (n+ 1) Γ (υ + 1)4nn!Γ (υ + n+ 1)∣∣∣∣∣≥ 1− 32 (v + 1)∞∑n=1(1v + 2)n−1=2v2 + v − 44 (v + 1)2By combining the above inequalities, it can be easily obtained∣∣∣∣zq′′υ (z)q′υ (z)∣∣∣∣ ≤ 3v + 62v2 + v − 4 , v > −1 +√334.Int. J. Anal. Appl. 16 (4) (2018) 4662. Convexity of Integral Operators Defined by Generalized Dini FunctionsThe main objective of this paper is to give convexity properties of integral operators involving Dinifunction. The main results are given as follows.Theorem 2.1. Let v1, . . . , vn >−5+√578 , where n ∈ N. Let qυi : U → C be defined asqυi (z) = 2υiΓ (υ + 1) z1−υi2((1− υi) Jυi(√z)+√zJ ′υi(√z)). (2.1)Suppose that v = min {v1, . . . , vn} and α1, . . . , αn be positive real numbers such that these numbers satisfythe following inequality0 ≤ 1− 3v + 64v2 + 5v − 2n∑i=1αi < 1.Then, the function Fv1,...,vn,α1,...,αn defined by (1.8), is in the class C (β), whereβ = 1− 3v + 64v2 + 5v − 2n∑i=1αi.Proof. We observe that qυi ,∀i = 1, 2, · · ·n are such that qυi (0) = q′υi (0) − 1 = 0. It is also clear thatFv1,...,vn,α1,...,αn ∈ A. That is Fv1,...,vn,α1,...,αn (0) = F ′v1,...,vn,α1,...,αn (0) − 1 = 0. On the other hand, it iseasy to see thatF ′v1,...,vn,α1,...,αn (z) =n∏i=1(qvi(z)z)αi. (2.2)Differentiating logarithmically, we getzF ′′v1,...,vn,α1,...,αn (z)F ′v1,...,vn,α1,...,αn (z)=n∑i=1αi(zq′vi (z)qvi (z)− 1). (2.3)This implies thatRe{1 +zF ′′v1,...,vn,α1,...,αn (z)F ′v1,...,vn,α1,...,αn (z)}=n∑i=1αiRe(zq′vi (z)qvi (z))+(1−n∑i=1αi).Now, by using the assertion (i) of Lemma 1.1 for each vi, where i = 1, 2, · · ·n, we obtainRe{1 +zF ′′v1,...,vn,α1,...,αn (z)F ′v1,...,vn,α1,...,αn (z)}≥n∑i=1αi(1− 3vi + 64v2i + 5vi − 2)+(1−n∑i=1αi)= 1−n∑i=1αi3vi + 64v2i + 5vi − 2.Consider the function φ :(−5+√578 ,∞)→ R defined asφ (x) =3x+ 64x2 + 5x− 2is decreasing function such that3vi + 64v2i + 5vi − 2≤ 3v + 64v2 + 5v − 2, ∀i = 1, 2, · · ·n.Int. J. Anal. Appl. 16 (4) (2018) 467ThereforeRe{1 +zF ′′v1,...,vn,α1,...,αn (z)F ′v1,...,vn,α1,...,αn (z)}> 1− 3v + 64v2 + 5v − 2n∑i=1αi.Since 0 ≤ 1− 3v+64v2+5v−2n∑i=1αi < 1, therefore Fv1,...,vn,α1,...,αn ∈ C (β), whereβ = 1− 3v + 64v2 + 5v − 2n∑i=1αi,which completes the proof. By setting α1 = α2 = · · · = αn = α in Theorem 2.1, we obtain the result given below.Corollary 2.1. Let v1, . . . , vn >−5+√578 , where n ∈ N. Let qυi : U → C be defined asqυi (z) = 2υiΓ (υ + 1) z1−υi2((1− υi) Jυi(√z)+√zJ ′υi(√z)). (2.4)Suppose that v = min {v1, . . . , vn} and α be positive real numbers such that these numbers satisfy the followinginequality0 ≤ 1− nα (3v + 6)4v2 + 5v − 2< 1.Then, the function Fv1,...,vn,α defined byFv1,...,vn,α (z) =z∫0n∏i=1(qvi(t)t)αdtis in the class C (β1), whereβ = 1− nα (3v + 6)4v2 + 5v − 2.The next theorem gives convexity properties of the integral operator defined in (1.9). The key tool in theproof is inequality (ii) of Lemma 1.1.Theorem 2.2. Let v1, . . . , vn >−1+√334 , where n ∈ N. Let qυi : U → C be defined asqυi (z) = 2υiΓ (υ + 1) z1−υi2((1− υi) Jυi(√z)+√zJ ′υi(√z)). (2.5)Suppose that v = min {v1, . . . , vn} and δ1, . . . , δn be positive real numbers such that these numbers satisfy thefollowing inequality0 ≤ 1− 3v + 62v2 + v − 4n∑i=1δi < 1.Then, the function Hv1,...,vn,δ1,...,δn defined by (1.9), is in the class C (γ), whereγ = 1− 3v + 62v2 + v − 4n∑i=1δi.Int. J. Anal. Appl. 16 (4) (2018) 468Proof. It can easily be observed that, the operator defined in (1.9) belongs to classA, that isHv1,...,vn,δ1,...,δn (0) =H ′v1,...,vn,δ1,...,δn (0)− 1 = 0. Differentiating (1.9), we haveH ′v1,...,vn,δ1,...,δn (z) =n∏i=1(q′vi(z))δi.Differentiating logarithmically, we obtainzH ′′v1,...,vn,δ1,...,δn (z)H ′v1,...,vn,δ1,...,δn (z)=n∑i=1δi(zq′′vi (z)q′vi (z)).This implies thatRe{1 +zH ′′v1,...,vn,δ1,...,δn (z)H ′v1,...,vn,δ1,...,δn (z)}= 1 +n∑i=1δiRe(zq′′vi (z)q′vi (z)).Now, by using the assertion (ii) of Lemma 1.1 for each vi, where i = 1, 2, · · ·n, we obtainRe{1 +zH ′′v1,...,vn,δ1,...,δn (z)H ′v1,...,vn,δ1,...,δn (z)}> 1−n∑i=1δi(3vi + 62v2i + vi − 4).Consider the function ϕ :(−1+√334 ,∞)→ R defined asϕ (x) =3x+ 62x2 + x− 4is decreasing function such that3vi + 62v2i + vi − 4≤ 3v + 62v2 + v − 4, ∀i = 1, 2, · · ·n.It follows thatRe{1 +zH ′′v1,...,vn,δ1,...,δn (z)H ′v1,...,vn,δ1,...,δn (z)}> 1− 3v + 62v2 + v − 4n∑i=1δi.Since0 ≤ 1− 3v + 62v2 + v − 4n∑i=1δi < 1,therefore Hv1,...,vn,δ1,...,δn ∈ C (γ), whereγ = 1− 3v + 62v2 + v − 4n∑i=1δi,which completes the proof. By setting δ1 = δ2 = δn = δ in Theorem 2.2, we obtain the result given below.Corollary 2.2. Let v1, . . . , vn >−1+√334 , where n ∈ N. Let qυi : U → C be defined asqυi (z) = 2υiΓ (υ + 1) z1−υi2((1− υi) Jυi(√z)+√zJ ′υi(√z)). (2.6)Suppose that v = min {v1, . . . , vn} and δ be positive real numbers such that these numbers satisfy the followinginequality0 ≤ 1− nδ (3v + 6)2v2 + v − 4< 1.Int. J. Anal. Appl. 16 (4) (2018) 469Then, the function Hv1,...,vn,δ defined asHv1,...,vn,δ (z) =z∫0n∏i=1(q′vi(t))δdtis in the class C (γ1), whereγ1 = 1−nδ (3v + 6)2v2 + v − 4.3. Some particular cases of Dini functionBy choosing v = 12 and v =32 in (1.2), we get the following forms of the normalized Dini functionq 12(z) =32√z(sin√z +√z cos√z),q 32(z) =32√z((z − 1) sin√z +√z cos√z).In particular, the results of the above mentioned theorems are given below.Corollary 3.1. Let v > −5+√578 and qυ : U → C be defined asqυ (z) = 2υΓ (υ + 1) z1−υ2((1− υ) Jυ(√z)+√zJ ′υ(√z)).Suppose that α be positive real number such that these numbers satisfy the following inequality0 ≤ 1− α (3v + 6)4v2 + 5v − 2< 1.Then, the function Fv,α defined byFv,α (z) =z∫0(qv(t)t)αdtis in the class C (β2), whereβ2 = 1−α (3v + 6)4v2 + 5v − 2.In particular,(i) if α ≤ 15, then the function F 12 ,α: U → C defined byF 12 ,α(z) =z∫0(32(sin√t+√t cos√t)√t)αdtis in the class C (β3), where β3 = 1− α15 .(ii)If α ≤ 2921 , then the function F 32 ,α : U → C defined byF 32 ,α(z) =z∫0(32((t− 1) sin√t+√t cos√t)t)αdtis in the class C (β4), where β4 = 1− 21α29 .Int. J. Anal. Appl. 16 (4) (2018) 470Corollary 3.2. Let v > −1+√334 and qυ : U → C be defined asqυ (z) = 2υΓ (υ + 1) z1−υ2((1− υ) Jυ(√z)+√zJ ′υ(√z)).Suppose that δ be positive real numbers such that these numbers satisfy the following inequality0 ≤ 1− δ (3v + 6)2v2 + v − 4< 1.Then, the function Hv,δ defined asHv,δ (z) =z∫0(q′v(t))δdtis in the class C (γ2), whereγ2 = 1−nδ (3v + 6)2v2 + v − 4.In particular,(i) if δ ≤ 421 , then the function H 32 ,α : U → C defined byH 32 ,α(z) =z∫0(q′32(t)δdtis in the class C (γ3), where γ3 = 1− 214 δ.References[1] Á. Baricz, Bessel transforms and Hardy space of generalized Bessel functions, Mathematica, 48(71) (2006), 127-136.[2] Á. Baricz, Generalized Bessel Functions of the First Kind, Lecture Notes in Mathematics Vol. 1994, SpringerVerlag, Berlin,2010.[3] Á. Baricz, E. Deniz and N. Yagmur, Close-to-convexity of normalized Dini functions, Math. Nech., 289 (2016), 1721-1726.[4] Á. Baricz, S. Ponnusamy and S. Singh, Modified Dini functions: monotonicity patterns and functional inequalities, ActaMath. Hungrica, 149(2016), 120-142.[5] Á. Baricz and R. Szász, The radius of convexity of normalized Bessel functions, Anal. Math., 41(3) (2015) 141-151.[6] D. Bansal and J. K. Prajapat. Certain geometric properties of the Mittag-Leffler functions, Complex Var. Ellipt. Equ.,61(3) (2016), 338-350.[7] E. Deniz, Ş. Gören and M. Çağlar, Starlikeness and convexity of the generalized Dini functions, AIP Conference Proceedings1833, 020004 (2017); doi: 10.1063/1.4981652.[8] E. Deniz, H. Orhan and H. M. Srivastava, Some sufficient conditions for univalence of certain families of integral operatorsinvolving generalized Bessel functions, Taiwanese J. Math. 15(2) (2011), 883-917.[9] M. Din, M. Raza and E. Deniz, Sufficient conditions for univalence of integral operators involving Dini Functions, (Sub-mitted).[10] M. Din, H. M. Srivastava, M. Raza, Univalence of integral operators involving generalized Struve functions, Hacettepe J.Math. Stat., Doi: 10.15672/HJMS.2017.485.[11] H. Orhan and N. Yagmur, Geometric properties of generalized Struve functions, In The International Congress in Honourof Professor Hari M. Srivastava, (2012), 23-26.Int. J. Anal. Appl. 16 (4) (2018) 471[12] H. M. Srivastava, B. A. Frasin and Virgil Pescar, Univalence of integral operators involving Mittag-Leffler functions, Appl.Math. Inf. Sci. 11(3) (2017), 635-641.[13] G. N. Watson. A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge,London and New York, 1944.

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Convexity of Integral Operators Involving Dini Functions

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