# Certain Geometric Properties of Generalized Dini Functions

Authors: Muhey U. Din; Mohsan Raza; Saqib Hussain; Maslina Darus
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We are mainly interested in some geometric properties for the combinations of generalized Bessel functions of the first kind and their derivatives known as Dini functions. In particular, we study the starlikeness of orderα, convexity of order α, and close-to-convexity of order ((1+α)/2) for normalized Dini function. We also study close-to-convexity with respect to certain star-like functions. Further, we obtain conditions on generalized Dini function to belong to the Hardy space Hp.


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## Body

## 1. Introduction

LetA be the class of functions f of the form(1)fz=z+∑n=2∞anzn,analytic in the open unit disc U=z:z<1 and S denote the class of all functions in A which are univalent in U. Let S∗α,Cα, Kα, and S∗~α denote the classes of star-like, convex, close-to-convex and strongly star-like functions of order α, respectively, and they are defined as (2)S∗α=f:f∈AandRezf′zfz>α,z∈U,α∈0,1,Cα=f:f∈AandRe1+zf′′zf′z>α,z∈U,α∈0,1,Kα=f:f∈AandRezf′zgz>α,z∈U,α∈0,1,g∈S∗,(3)S∗~α=f:f∈Aandargzf′zfz<απ2,z∈U,α∈0,1.It is clear that (4)S∗~1=S∗0=S∗,C0=C,K0=K.Iff and g are analytic functions, then the function f is said to be subordinate to g, written as f(z)≺g(z), if there exist a Schwarz function w which is analytic with w(0)=0 and w<1 such that f(z)=g(w(z)). Furthermore, if the function g is univalent in U then we have the following equivalent relation: (5)fz≺gz⟺f0=g0,fU⊂gU.Special functions have great importance in pure and applied mathematics. The wide use of these functions has attracted many researchers to work on the different directions. Geometric properties of special functions such as hypergeometric functions, Bessel functions, Struve functions, Mittag-Leffler functions, Wright functions, and some other related functions are an ongoing part of research in geometric function theory. We refer to some geometric properties of these functions [1–10] and references therein.Consider the second-order homogeneous differential equation(6)z2w′′z+bzw′z+cz2-v2+1-bvwz=0,where b,c, and v are complex numbers. The particular solution of the homogenous differential equation (6) is called the generalized Bessel functions of the first kind of order v. It is defined as(7)wv,b,cz=∑n=0∞-cnz/22n+vn!Γv+n+b+1/2,where Γ. denotes the gamma function. The function wv,b,c unifies the Bessel, modified Bessel, and spherical Bessel functions.Special cases ( 1 )
Forb=c=1, we have the Bessel functions of first kind of order v, defined as(8)Jvz=∑n=0∞-1nz/22n+vn!Γv+n+1. ( 2 )
Forb=1,c=-1, we obtain the modified Bessel functions of first kind of order v whose series form is given as(9)Ivz=∑n=0∞z/22n+vn!Γv+n+1. ( 3 )
Forb=2,c=1, we have the spherical Bessel functions of first kind of order v, given as(10)jvz=π2∑n=0∞z/22n+vn!Γv+n+3/2. For more details about these functions, see [5, 11]. Recently, Deniz et al. [12] studied the function φv,b,c which is defined by the relation as (11)φv,b,cz=2vΓv+b+12z1-v/2wv,b,cz1/2, whereb,c,v∈C. Using the well-known Pochhammer symbol (12)γn=Γγ+nΓγ=1,n=0,γ∈C\0,γγ+1…γ+n-1,n∈N,γ∈C, we obtain the series form of the functionφv,b,c as(13)φv,b,cz=z+∑n=1∞-cn4nknn!zn+1, wherek=v+b+1/2≠0,-1,-2,⋯.Bessel functions are indispensable in many branches of mathematics and applied mathematics. Thus, it is important to study their properties in many aspects. We consider the generalized Dini functionsdυ,a,b,c:U→C defined as(14)dυ,a,b,cz=2υaΓυ+b+12z1-υ/2a-υφv,b,cz+zφv,b,c′z=z+∑n=1∞-cn2n+aΓυ+1a4nn!Γυ+n+1zn+1=z+∑n=1∞-cn2n+a4nn!aknzn+1,where a∈R+,b∈R,c∈C, and v∈R. For a=b=c=1, we obtain the normalized Dini function of order v of the form (15)dυz=z+∑n=1∞-1n2n+1Γυ+14nn!Γυ+n+1zn+1=z+∑n=1∞-1n2n+1Γk4nn!Γk+nzn+1=z+∑n=1∞-1n2n+14nn!knzn+1. By putting v=1/2 and v=3/2, we obtain the following particular cases of normalized Dini functions (16)d1/2z=zcos⁡z,d3/2z=3cos⁡z-3z-2sin⁡z2z.Recently Baricz et al. [13] studied the close-to-convexity of Dini functions and some monotonicity properties and functional inequalities for the modified Dini function are discussed in [5]. Further some geometric properties of Dini functions are studied in [14].This paper studies the Dini functiondv,a,b,c given by the power series (14). We determine the conditions on parameters that ensure the Dini function to be star-like of order α, convex of order α, and close-to-convex of order ((1+α)/2). We also study the convexity in the domain U1/2=z:z<1/2. Sufficient conditions on univalency of an integral operator defined by Dini function are also studied. We find the conditions on normalized Dini function to belong to the Hardy space Hp.To prove our main results, we need the following lemmas.Lemma 1 (see [15]).
Iff∈A satisfy f′(z)-1<1 for each z∈U, then f is convex in U1/2=z:z<1/2.Lemma 2 (see [16]).
Letβ∈C with Re(β)>0, and c∈C with c≤1, c≠-1. If h∈A satisfies (17)cz2β+1-z2βzh′′zβh′z≤1,z∈U,then the integral operator (18)Cβz=β∫0ztβ-1h′tdt1/β,z∈U, is analytic and univalent in U.Lemma 3 (see [17]).
LetG(z) be convex and univalent in the open unit disc with condition G(0)=1. Let F(z) be analytic in the open unit disc with condition F(0)=1 and F≺G in the open unit disc. Then, ∀n∈N∪{0}, we obtain (19)n+1z-1-n∫0ztnFtdt≺n+1z-1-n∫0ztnGtdt.Lemma 4 (see [18]).
Iff∈A satisfies the inequality (20)zf′′z<1-α4,z∈U,0≤α<1, then (21)Ref′z>1+α4,z∈U,0≤α<1.Lemma 5 (see [19]).
If the functionfz=z+a2z2+…+anzn+… is analytic in U and in addition 1≥2a2≥…≥nan…≥0 or 1≤2a2≤…≤nan…≤2, then f is close-to-convex function with respect to the convex function z→-log1-z. Moreover, if the odd function g(z)=z+b3z3+…+b2n-1z2n-1+… is analytic in U and if 1≥3b3≥…≥(2n+1)b2n+1…≥0 or 1≤3b3≤…≤(2n+1)b2n+1…≤2, then g is univalent in U.


## 2. Geometric Properties of Normalized Dini Functions

Theorem 6.
Leta∈R+, b∈R, c∈C, and v∈R with α∈0,1 and z∈U. Then the following assertions are true:(i)
If(2c+ac4k+1-c+4c2+ac2)/((4(k+1)-c)(4ak-2c-ac)-c22+a) < 1-α, then dv,a,b,c∈S∗(α).(ii)
If((4(k+1)-c)(4c+2ac)+9c2+2c2(2+a)+ac2)/((4(k+1)-c)(4ak − 4c-2ac)-4c2-2c2(2+a)-ac2) < 1-α, then dv,a,b,c∈C(α).(iii)
If((4(k+1)-c)(4c+2ac)+9c2+2c2(2+a)+ac2)/(4ak(4k+1-c) < (1-α)/4, then dv,a,b,c∈K((1+α)/2).(iv)
If(2ck+1+ack+1)/(ak(4(k+1)-c))<1-α, then dv,a,b,c/z∈P(α).Proof.
We use the inequalityzdv,a,b,c′(z)/dv,a,b,c(z)-1<1-α to prove the starlikeness of order α for the function dv,a,b,c. So by using the well-known triangle inequality (22)z1+z2≤z1+z2, with the equality (23)Γv+1Γv+n+1=1v+1v+2…v+n=1v+1n=1kn,n∈N, and the inequalities (24)2n-1!k+1n-1≥nk+1n-1,n-1!k+1n-1≥k+1n-1,n∈N, we obtain(25)dv,a,b,c′z-dv,a,b,czz≤∑n=1∞n2n+acna4nn!kn≤c2ak+c4k+cak∑n=2∞c4k+1n-1+c4k∑n=2∞c4k+1n-1=2c+ac4k+1-c+c24+a4ak4k+1-c. Furthermore, if we use reverse triangle inequality (26)z1+z2≥z1-z2, and the inequalities (27)n!k+1n-1≥nk+1n-1,n!k+1n-1≥k+1n-1,n∈N,then we get(28)dv,a,b,c′zz=1+∑n=1∞-cn2n+aa4nn!kn)zn≥1-c2ak+c4k+c2ak∑n=2∞c4k+1n-1+c4k∑n=2∞c4k+1n-1=-c22+a+4k+1-c4ak-2c-ac4ak4k+1-c By combining inequalities (25) and (28), we get (29)zdv,a,b,c′zdv,a,b,cz-1≤2c+ac4k+1-c+c24+a-c22+a+4k+1-c4ak-2c-ac<1-α.So, dv,a,b,c is star-like function of order α, where 0≤α<1-(2c+ac(4(k+1)-c) + c24+a)/(4k+1-c4ak-2c-ac − c22+a).
(ii) To prove the convexity of orderα of function dv,a,b,c, we have to show that zdv,a,b,c′′(z)/dv,a,b,c′(z)<1-α. By using the well-known triangle inequality (30)z1+z2≤z1+z2, with equality (31)Γv+1Γv+n+1=1v+1v+2…v+n=1v+1n=1kn,n∈N, and the inequalities (32)9n-1!k+1n-1≥2n2k+1n-1,2n-1!k+1n-1≥nk+1n-1,n-1!k+1n-1≥k+1n-1,n∈N, we have(33)zdv,a,b,c′′z=∑n=1∞-an2n+an+1nΓv+1a4nn!Γv+n+1zn≤∑n=1∞2n+an+1a4nn!kn≤cak+c2k+9c24ak4k+1-c+2+ac22ak4k+1-c=4k+1-c4c+2ac+9c2+2c22+a+ac24ak(4k+1-c.Moreover, if we use reverse triangle inequality (34)z1+z2≥z1-z2, with equality (35)Γv+1Γv+n+1=1v+1v+2…v+n=1v+1n,n∈N, and the inequalities (36)2n-1!k+1n-1≥nk+1n-1,2n!k+1n-1≥nk+1n-1,n!k+1n-1≥k+1n-1,n∈N, we get(37)dv,a,b,c′z≥1-∑n=1∞2n+an+1a4nn!cn≤1-cak-c2k-c2ak4k+1-c-2+ac22ak4k+1-c-c24k4k+1-c=4k+1-c4ak-4c-2ac-4c2-2c22+a-ac24ak(4k+1-c.By combining inequalities (33) and (37), we have(38)zdv,a,b,c′′zdv,a,b,c′z≤4k+1-c4c+2ac+9c2+2c22+a+ac24k+1-c4ak-4c-2ac-4c2-2c22+a-ac2<1-α. This implies that dv,a,b,c is convex function of order α, where 0≤α<1-(4k+1-c4c+2ac + 9c2+2c2(2+a)+ac2)/((4(k+1) − c)(4ak-4c-2ac) − 4c2-2c2(2+a)-ac2).
(iii) Using inequality (33) and Lemma 4, we have (39)zdv,a,b,c′′z≤4k+1-c4c+2ac+9c2+2c22+a+ac24ak(4k+1-c<1-α4, where 0≤α<1-4(((4(k+1)-c)(4c+2ac)+9c2+2c2(2+a)+ac2)/(4ak(4(k+1)-c)). This shows that dv,a,b,c∈K((1+α)/2). Therefore Redv,a,b,c′(z)>(1+α)/2.
(iv) To prove thatdv,a,b,c/z∈P(α), we have to show that g(z)-1<1, where g(z)=(dv,a,b,c(z)/(z-α))/(1-α). By using the well-known triangle inequality (40)z1+z2≤z1+z2, with the equality (41)Γv+1Γv+n+1=1v+1v+2…v+n=1v+1n,n∈N, and the inequality (42)n-1!k+1n-1≥k+1n-1,n∈N, we have (43)gz-1=11-α∑n=1∞-cn2n+aa4nn!knzn≤11-α2c4ak∑n=1∞c4k+1n-1+11-αc4k∑n=1∞c4k+1n-1=11-α2ck+1+ack+1ak4k+1-c. Therefore, dv,a,b,c/z∈P(α) for 0<α<1-(2ck+1+ack+1)/(ak(4(k+1)-c)).Corollary 7.
By using Theorem6 assertions (i) and (iv), we get the following corollary:(i)
If0≤α<α1 where α1≃0.333333333…, then d1/2∈S∗(α).(ii)
If0≤α<α2 where α2≃0.7727272727…, then d3/2∈S∗(α).(ii)
If0≤α<α3 where α3≃0.44444444…, then d1/2/z∈P(α).(iii)
If0≤α<α4 where α4≃0.67692307…, then d3/2/z∈P(α).Puttingα=0 in Theorem 6, we have the following results.Corollary 8.
Leta∈R+, b∈R, c∈C, and v∈R. Then the following assertions are true.
( i ) If k>(8c(a+1)-4a(4-c) + (8ca+1-4a4-c2 + 64a2c4-ca+1+2c2a+3+2c)1/2)/32a, then dv,a,b,c∈S∗.
( i i ) If k>(4(8c+4ac)-4a(4-c) + (48c+4ac-4a4-c2 + 64a4-c8c+4ac+c26a+21)1/2)/32a, then dv,a,b,c∈C.
( i i i ) If k>(8c(a+2)-a(4-c) + (8ca+2-a4-c2 + 16a2ca+24-c+c23a+13)1/2)/8a, then dv,a,b,c∈K(1/2).
( i v ) If k>(c(a+2)-a(4-c)+c|(a+2)-a(4-c)2+16aca+2)/8a, then dv,a,b,c/z∈P.Corollary 9.
By puttinga=b=c=1 and k=v+1 in Corollary 8, we obtain the following results:
( i ) If v>0.453768227, then dv∈S∗.
( i i ) If v>2.406035949, then dv∈C.
( i i i ) If v>5.548088707, then dv∈K(1/2).
( i v ) If v>-0.8660254038, then dv/z∈P.Remark 10.
It is easy to see that our results improve the results of [14] as special cases of parameters. Some particular cases regarding Corollary 9 are given below.Corollary 11.
( i ) If v=1/2, then d1/2(z)=zcosz∈S∗.
( i i ) If v=3/2, then d3/2(z)=3cosz-(3(z-2)sinz)/2z∈S∗.
( i i i ) If v=1/2, then (d1/2/z)(z)=cosz∈P.
( i v ) If v=3/2, then d3/2(z)/z=(6zcosz-3z-2sinz)/2z3/2∈P.Theorem 12.
Leta∈R+, b∈R, c∈C, and v∈R. If (4k+1-cca+4 + c2a+6+4ack+1)/(4ak4k+1-c) < 1, then dυ,a,b,c is convex in U1/2.Proof.
By using the well-known triangle inequality(44)z1+z2≤z1+z2, with the equality Γυ+1/Γυ+n+1=1/(υ+1υ+2…υ+n)=1/υ+1n, n∈N, and the inequalities (45)2n!k+1n-1≥n2k+1n-1,n∈N,n!k+1n-1≥nk+1n-1,n∈N,n!k+1n-1≥k+1n-1,n∈N, we obtain (46)dv,a,b,c′z-1≤∑n=1∞2n2+na+2+aa4nn!kncn≤c2ak+cak∑n=2∞c4k+1n-1+a+2c4ak+a+2c4ak∑n=2∞c4k+1n-1+c4k∑n=1∞c4k+1n-1=4k+1-cca+4+c2a+6+4ack+14ak4k+1-c. In view of Lemma 1, dυ,a,b,c is convex in U1/2, if (4k+1-cca+4+c2a+6+4ack+1)/(4ak4k+1-c)<1, but this is true under the hypothesis.Remark 13.
When we puta=b=c=1, our results are better than [14].
Consider the integral operatorFβ:U→C, where β∈C, β≠0, (47)Fβz=β∫0ztβ-2dυ,a,b,ct1/β,z∈U. Here Fβ∈A. In the next theorem, we obtain the conditions so that Fβ is univalent in U.Theorem 14.
Leta∈R+, b∈R, c∈C, and v∈R. Let (2c+ac(4(k+1)-c)+4c2+ac2)/(-c22+a+4k+1-c4ak-2c-ac)<1 and suppose that M is a positive real number such that dυ,a,b,cz≤M in the open unit disc. If (48)β-1+2c+ac4k+1-c+4c2+ac24k+1-c4ak-2c-ac-c22+a+Mβ≤1, then Fβ is univalent in U.Proof.
A calculation gives us(49)zFβ′′zFβ′z=zdυ,a,b,c′zdυ,a,b,cz+zβ-1βdυ,a,b,cz+β-2,z∈U.Since dυ,a,b,c∈A, then, by the Schwarz Lemma, triangle inequality, and (29), we obtain (50)1-z2zFβ′′zFβ′z≤1-z2β-1+zdυ,a,b,c′zdυ,a,b,cz-1+zRββdυ,a,b,czz≤1-z2β-1+2c+ac4k+1-c+4c2+ac2-c22+a+4k+1-c4ak-2c-ac+Mβ≤1.This shows that the given integral operator satisfies Becker’s criterion for univalence, and hence Fβ is univalent in U.


## 3. Strong Starlikeness

In this section, we are mainly interested about some sufficient conditions under which the normalized Dini function belongs to the class of strong starlikeness of orderα.Theorem 15.
Leta∈R+, b∈R, c∈C, and v∈R. If (4k+1-cca+4+c2a+6+4ack+1)/(4ak4k+1-c)<1, then dυ,a,b,c∈S∗~α, where (51)α=2πarcsin⁡ϰ1-ϰ24+ϰ41-ϰ2, and ϰ=(4k+1-cca+4+c2a+6+4ack+1)/(4ak4k+1-c).Proof.
By using the well-known triangle inequality(52)z1+z2≤z1+z2, with the equality Γυ+1/Γυ+n+1=1/(υ+1υ+2…υ+n)=1/υ+1n, n∈N, and the inequalities (53)2n!k+1n-1≥n2k+1n-1,n∈N,n!k+1n-1≥nk+1n-1,n∈N,n!k+1n-1≥k+1n-1,n∈N, we obtain(54)dv,a,b,c′z-1≤∑n=1∞2n2+na+2+aa4nn!kncn≤c2ak+cak∑n=2∞c4k+1n-1+a+2c4ak+a+2c4ak∑n=2∞c4k+1n-1+c4k∑n=1∞c4k+1n-1(55)=4k+1-cca+4+c2a+6+4ack+14ak4k+1-c=ϰ.For 0<ϰ≤1 and from (54), we concluded that(56)dv,a,b,c′≺1+ϰz⟹arg⁡dv,a,b,c′z<arcsin⁡ϰ.With the help of Lemma 3, take n=0 with F(z)=dv,a,b,c′(z) and G(z)=1+ϰz, and we get(57)dv,a,b,c′z≺1+ϰ2z. As a result(58)arg⁡dv,a,b,czz<arcsin⁡ϰ2. By using (56) and (58), we obtain (59)arg⁡zdv,a,b,c′zdv,a,b,cz=arg⁡zdv,a,b,cz+arg⁡dv,a,b,c′z≤arg⁡zdv,a,b,cz+arg⁡dv,a,b,c′z<arcsin⁡ϰ2+arcsin⁡ϰ=arcsin⁡ϰ1-ϰ24+ϰ41-ϰ2, which implies that dυ,a,b,c∈S∗~α for α=(2/π)arcsinϰ1-ϰ2/4+(ϰ/4)1-ϰ2.Corollary 16.
Fork=k∗, where k∗ is the positive root of 2k2-35k-25=0, then dυ,a,b,cz∈S∗.


## 4. Close-to-Convexity with respect to Certain Functions

Recently many authors discuss the close-to-convexity of some special functions with respect to certain functions. Here, we are also interested in the close-to-convexity of normalized Dini function.Theorem 17.
Leta∈R+, b∈R, c∈R, and v∈R. If Γ(v+n+1)≥-c2n+aΓv+n/(2(n-1)+a), then dv,a,b,c is close-to-convex with respect to -log(1-z).Proof.
Set(60)fz=z+∑n=2∞bn-1zn. To prove that f is close-to-convex with respect to the function -log1-z,we use Lemma 5. Therefore, we have to prove that nbn-1n≥2 is a decreasing sequence. After some computations, we obtain (61)nbn-1-n+1bn=n-cn-12n-1+aΓv+1a4n-1n-1!Γv+n-n+1-cn2n+aΓv+1a4nn!Γv+n+1=-cn-1Γv+1a4n-1n-1!4n2Γv+n+12n-1+a+cn+12n+aΓv+n4nΓv+n+1Γv+n>0. By using the conditions on parameters, we easily observe that nbn-1-(n+1)bn>0 for all n≥2, and thus nbn-1n≥2 is a decreasing sequence. By Lemma 5 it follows that f is close-to-convex with respect to the function -log⁡1-z.Theorem 18.
Leta∈R+, b∈R, c∈R, and v∈R. If Γ(v+n+1)≥-c(2n+a)Γ(v+n)/(2(n-1)+a), then dv,a,b,c is close-to-convex with respect to 1/2log1+z/1-z.Proof.
Set(62)fz=z+∑n=2∞B2n-1z2n-1. Here B2n-1=bn-1=(-c)n-1(2(n-1)+a)Γ(v+1)/a4n-1(n-1)!Γ(v+n+1), and therefore we have b1=-c(2+a)/4ak and B2n-1>0 for all n≥2. To prove our main result we will prove that (2n-1)bn-1n≥2 is a decreasing sequence. After some computations, we obtain(63)2n-1bn-1-2n+1bn=2n-1-cn-12n-1+aΓv+1a4n-1n-1!Γv+n-2n+1-cn2n+aΓv+1a4nn!Γv+n+1,=-cn-1Γv+1a4n-1n-1!4n2n-1Γv+n+12n-1+a+c2n+12n+aΓv+n4nΓv+n+1Γv+n>0. By using the conditions on parameters we easily observe that (2n-1)bn-1-(2n+1)bn>0 for all n≥2, and thus (2n-1)bn-1n≥2 is a decreasing sequence. By Lemma 5 it follows that f(z) is close-to-convex with respect to the function 1/2log⁡1+z/1-z.


## 5. Hardy Spaces of Dini Function

LetH denote the class of all analytic functions in the open unit disk U=z:z<1 and H∞ denote the space of all bounded functions on H. This is a Banach algebra with respect to the norm (64)f∞=supz∈Ufz. We denote Hp,0<p<∞ for the space of all functions f∈Hsuch that fp admits a harmonic majorant. Hp is a Banach space if the norm of f is defined to be p-th root of the least harmonic majorant of fp for some fixed z∈U. Another equivalent definition of norm is given as follows: Let f∈H, and set (65)
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=
∞
. Then the function f∈Hp if Mpr,f is bounded for all r∈0,1. It is clear that (66)H∞⊂Hq⊂Hp,0<q<p<∞. For some details see [20]. It is also known [20] that, for Ref′z>0 in U, then (67)f′∈Hq,q<1,f∈Hq/1-q,0<q<1. Hardy spaces of hypergeometric functions are recently studied by Ponnusamy [21]. Baricz [2] uses the idea of Ponnusamy and found the Hardy spaces of Bessel functions while Yag˘mur and Orhan [10] studied the same problem for generalized Struve functions. Similarly Yag˘mur [22] studied the problem for Lommel functions and Raza et al.[9] studied the same problem for Wright functions.To prove our main results we need the following lemmas.Lemma 19 (see [23]).
P 0 α ∗ P 0 β ⊂ P 0 ( γ ), where γ=1-2(1-α)(1-β) with α,β<1 and the value of γ is best possible.Lemma 20 (see [24]).
Forα,β<1 and γ=1-2(1-α)(1-β), we have R0α∗R0β⊂R0(γ) or equivalently P0α∗P0β⊂P0(γ).Lemma 21 (see [25]).
If the functionf, convex of order α, where α∈0,1, is not of the form(68)fz=ζ+η1-zeiγ2α-1,α≠12,ζ+ηlog⁡1-zeiγ,α≠12, for some complex numbers ζ and η and for some real number γ then the following statements hold:
( i ) There exists δ=δ(f)>0, such that f′∈Hδ+1/2(1-α).
( i i ) If α∈1,1/2, then there exists τ=τf>0, such that f∈Hτ+1/(1-2α).
( i i i ) If α≥1/2, then f∈H∞.The lemma defined below also plays an important role to find our main results.Theorem 22.
Letα∈0,1 and ((4(k+1)-c)(4c+2ac) + 9c2+2c2(2+a)+ac2)/((4(k+1)-c)(4ak-4c-2ac) − 4c2-2c2(2+a)-ac2) < 1-α. Then( i )  dv,a,b,c∈H1/1-2α for α∈0,1/2;( i i )  dv,a,b,c∈H∞ for α≥1/2.Proof.
By using the definition of Hypergeometric function(69)F12a,b,c;z=∑n=0∞anbncnznn!, we have (70)ζ+ηz1-zeiγ1-α=ζ+ηzF121,1-2α,1;zeiγ=ζ+η∑n=0∞1-2αnn!eiγnzn+1, for ζ, η∈C, and α≠1/2 and for γ∈R. On the other hand (71)ζ+ηlog⁡1-zeiγ=ζ-ηzF121,1,2;zeiγ=ζ-η∑n=0∞1n+1eiγnzn+1. This implies that dv,a,b,c is not of the form ζ+ηz(1-zeiγ)2α-1 for α≠1/2 and ζ+ηlog(1-zeiγ) for α=1/2, respectively. Also, from part (ii) of Theorem 6, dv,a,b,c is convex of order α. Hence, by using Lemma 21, we have required result.Theorem 23.
Letv>-0.8660254038 and f∈R. Then the convolution dv∗f is in H∞∩R.Proof.
Leth(z)=dv∗f(z). Then it is clear that h′(z)=dv/z∗f′(z). Using Corollary 9 of part (iv), we have dv/z∈P. Since f∈R, therefore by using Lemma 19  h∈R. It is also clear that dv/z is an entire function and therefor h is entire. This implies that h is bounded. Hence, we have the required result.Theorem 24.
Letv∈R with (2ck+1+ack+1)/ak(4(k+1)-c)<1-α, and then dv,a,b,c/z∈P(α), α∈0,1, and z∈U. If f∈R(β), with β<1, then dv,a,b,c∗f∈R(γ), where γ=1-2(1-α)(1-β).Proof.
Leth(z)=dv,a,b,c(z)∗f(z), and then h′(z)=dv,a,b,cz/z∗f′(z). Now, from Theorem 6 of part (iv), we have dv,a,b,c/z∈P(α). By using Lemma 19 and the fact that f′∈P(β), we have h′(z)∈Pγ, where γ=1-2(1-α)(1-β). Consequently, we have h∈R(γ).Corollary 25.
Let0≤α<1, (2ck+1+ack+1)/ak(4(k+1)-c)<1-α, and then dv,a,b,c/z∈P(α). Iff∈R(β), β=(1-2α)(2-2α), then dv,a,b,c∗f∈R(0).Corollary 26.
Letv>-0.8660254038. Iff∈R(1/2), then dv∗f∈R(0).By using Theorem23 and Corollary 25, then we have the following corollary.Corollary 27.
Iff∈R, then the convolutions d1/2∗f and d3/2∗f are in H∞∩R. Moreover, if f∈R(1/2), then d1/2∗f and d3/2∗f∈R(0).



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*Source: 2684023-2018-07-11.xml*