JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2016/1896154 1896154 Research Article Certain Geometric Properties of Normalized Wright Functions http://orcid.org/0000-0001-7466-7930 Raza Mohsan 1 Din Muhey U 1 http://orcid.org/0000-0001-8940-0569 Malik Sarfraz Nawaz 2 Petrusel Adrian 1 Department of Mathematics Government College University Faisalabad Faisalabad Pakistan gcuf.edu.pk 2 Department of Mathematics COMSATS Institute of Information Technology Wah Cantt Pakistan comsats.edu.pk 2016 27122016 2016 15 10 2016 04 12 2016 27122016 2016 Copyright © 2016 Mohsan Raza et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this article, we find some geometric properties like starlikeness, convexity of order α, close-to-convexity of order 1+α/2, and close-to-convexity of normalized Wright functions with respect to the certain functions. The sufficient conditions for the normalized Wright functions belonging to the classes Tγα and Lγα are the part of our investigations. We also obtain the conditions on normalized Wright function to belong to the Hardy space Hp.

COMSATS Institute of Information Technology 3-64/IPF-SRG/CIIT/Wah/14/765
1. Introduction and Preliminaries

Let H 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 Banach algebra with respect to the norm (1)f=supzUfz.We denote Hp, 0<p<, for the space of all functions fH such that fp admits a harmonic majorant. Hp is a Banach space if the norm of f is defined to be pth root of the least harmonic majorant of fp for some fixed zU. Another equivalent definition of norm is given as follows: let fH, and set (2) M p r , f = 1 2 π 0 2 π f r e i θ p d θ 1 / p , 0 < p < , max f z : z r , p = . Then the function fHp if Mpr,f is bounded for all r0,1. It is clear that (3)HHqHp,0<q<p<.For some details, see . It is also known  that Refz>0 in U, and then (4)fHq,q<1,fHq/1-q,0<q<1.Let A be the class of functions f of the form (5)fz=z+n=2anzn,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 Sα, Cα, and Kα denote the classes of starlike, convex, and close-to-convex functions of order α, respectively, and they are defined as(6)Sα=f:fA,Rezfzfz>α,zU,α0,1,Cα=f:fA,Re1+zfzfz>α,zU,α0,1,Kα=f:fA,Rezfzgz>α,zU,α0,1,gS.It is clear that (7)S0=S,C0=C,K0=Kare the classes of starlike, convex, and close-to-convex functions, respectively. Also consider the subclasses Tγα and Lγα of A, defined by the following relations:(8)Tγα=fA:Rezfz+γz2fz1-γfz+γzfz>α,Lγα=fA:Reγz3fz+1+2γz2fz+zfzzfz+γz2fz>α,where 0α,γ<1. The purpose of these subclasses is that when we put γ=0 in (8), we get T0α=Sα and L0α=Cα. The sufficient coefficient conditions by which a function fA as defined in (5) belongs to the classes Tγα and Lγα are(9)n=2nγ-γ+1n-αan1-α,(10)n=2nnγ-γ+1n-αan1-α,respectively. For some details about these classes, see [2, 3]. Recently, Baricz  introduced the classes(11)Pηα=p:pH,p0=1,Reeiηpz-α>0,zU,α0,1,ηR,Rηα=f:fA,Reeiηfz-α>0,zU,α0,1,ηR.For η=0,  we denote the classes P0α and R0α by Pα and Rα, respectively. Also for η=0 and α=0, we have the classes P and R.

Let fA given by (5) and gA given by(12)gz=z+n=2bnzn,and then Hadamard product (or convolution) of f and g is defined as(13)fgz=z+n=2anbnznzU.Recently, Prajapat  studied some geometric properties of Wright function(14)Wλ,μz=n=0znn!Γλn+μ,λ>-1,μC.This series is absolutely convergent in C for λ>-1 and absolutely convergent in open unit disc U for λ=-1. Furthermore this function is entire. The Wright functions were introduced by Wright  and have been used in the asymptotic theory of partitions, in the theory of integral transforms of the Hankel type and in Mikusinski operational calculus. Recently, Wright functions have been found in the solution of partial differential equations of fractional order. It was found that the corresponding Green functions can be represented in terms of the Wright function [7, 8]. For positive rational number λ, the Wright function can be represented in terms of generalized hypergeometric function. For some details, see [9, Section 2.1]. In particular, the function W1,v+1(-z2/4) can be expressed in terms of the Bessel functions Jv, given as(15)Jvz=z22W1,v+1-z24=n=0-1nz/22n+vn!Γn+v+1.The Wright function generalizes various functions like Array function, Whittaker function, entire auxiliary functions, and so forth. For more details, we refer to . Prajapat  discussed some geometric properties of the Wright functions,(16)Wλ,μz=n=0Γμn!Γλn+μzn,λ>-1,μ>0,zU,and their normalization of the form(17)zWλ,μz=Wλ,μz=z+n=1Γμn!Γλn+μzn+1,λ>-1,μ>0,zU,where λ+μ>0. 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). We refer for some geometric properties of special functions like hypergeometric functions [10, 11], Bessel functions [4, 1214], and Struve functions [15, 16].

We need the following results to prove our results.

Lemma 1 (see [<xref ref-type="bibr" rid="B15">17</xref>]).

If    fA satisfies the inequality(18)zfz<1-α4,zU,0α<1,then(19)Refz>1+α4,zU,0α<1.

Lemma 2 (see [<xref ref-type="bibr" rid="B16">18</xref>]).

If the function fz=z+a2z2++anzn+ is analytic in U and in addition 12a2nan0 or 12a2nan2, then f(z) 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 13b3(2n+1)b2n+10 or 13b3(2n+1)b2n+12, then g(z) is univalent in U.

Lemma 3 (see [<xref ref-type="bibr" rid="B25">19</xref>]).

P 0 α P 0 β P 0 γ , where γ=1-21-α1-β with α,β<1 and the value of γ is the best possible.

Lemma 4 (see [<xref ref-type="bibr" rid="B18">20</xref>]).

For α,β<1    and γ=1-21-α1-β, we have R0αR0βR0γ or equivalently PαP0βP0γ.

Lemma 5 (see [<xref ref-type="bibr" rid="B8">21</xref>]).

If the function f, convex of order α, where α0,1, is not of the form (20) f z = k + d z 1 - z e i γ 2 α - 1 , α 1 2 , k + d log 1 - z e i γ , α = 1 2 for some complex numbers k and d and for some real number γ, then the following statements hold:

There exists δ=δf>0 such that fHδ+1/21-α.

If α0,1/2, then there exists τ=τf>0 such that fHτ+1/1-2α.

If α1/2, then fH.

2. Main Results Theorem 6.

Let λ,μR with α0,1 and zU. Then the following assertions are true:

If λ1 and μ>2-α+5α2-16α+12/21-α, then Wλ,μSα.

If λ1 and μ>4-α+5α2-28α+32/21-α, then Wλ,μCα.

If λ1 and μ>6+48-12α/1-α, then Wλ,μK(1+α)/2.

If λ1 and μ>1+5-4α/21-α, then Wλ,μ/zPα.

Proof.

(i) To prove that Wλ,μSα,  we have to show that zWλ,μ(z)/Wλ,μ(z)-1<1-α. By using the well-known triangle inequality(21)z1+z2z1+z2with the inequality Γμ+nΓμ+nλ, nN, which is equivalent to Γμ/Γλn+μ1/μμ+1(μ+n-1)=1/μn, nN, and the inequality(22)n!μ+1n-1nμ+1n-1,nN,we obtain(23)Wλ,μz-Wλ,μzzn=1nΓμn!Γλn+μn=1nΓμn!Γn+μλ11μn=11μ+1n-1=μ+1μ2,μ>0,zU.Also consider (24)Wλ,μzz=1+n=1Γμn!Γλn+μzn.Since Γn+μΓλn+μ, therefore by using the reverse triangle inequality (25)z1-z2z1+z2and the inequality, n!μ+1n-1μ+1n-1, nN, we get(26)Wλ,μzz=1+n=1Γμn!Γλn+μzn1-n=11n!μn1-1μn=11μ+1n-1=μ2-μ-1μ2,μ>0.By combining (23) and (26), we get(27)zWλ,μzWλ,μz-1μ+1μ2-μ-1<1-α.So Wλ,μ(z) is starlike function of order α, where 0α<1-(μ+1)/(μ2-μ-1).

(ii) To prove that Wλ,μCα,  we have to show that zWλ,μ(z)/Wλ,μ(z)<1-α. By using the well-known triangle inequality(28)z1+z2z1+z2with the inequality Γμ+nΓμ+nλ, nN, which is equivalent to Γμ/Γλn+μ1/μμ+1μ+n-1=1/μn, nN, and the inequalities(29)n!nn+12n,μ+1n-1μ+1n-1,nN,we have (30)zWλ,μz=n=1Γμnn+1n!Γλn+μznn=1nn+12nΓμnn+1Γλn+μ2μn=12μ+1n-1=2μ+1μμ-1,μ>1.Since Γn+μΓλn+μ, therefore by using the reverse triangle inequality (31)z1-z2z1+z2and the inequality, n!μ+1n-1nμ+1n-1, nN, we get(32)Wλ,μz1-n=1nΓμn!Γn+μ=1-1μn=1nn!μ+1n-11-1μn=11μ+1n-1=μ2-μ-1μ2,μ>0.Combining (30) and (32), we have(33)zWλ,μzWλ,μz3μ+1μ2-μ-1<1-α.This implies that Wλ,μ is convex function of order α, where 0α<1-3μ+1/μ2-μ-1.

(iii) Using inequality (30) and Lemma 1, we have(34)zWλ,μz3μ+1μ21-α4,where 0α<1-12μ+1/μ2 and μ>6+48-12α/1-α. This shows that Wλ,μK(1+α)/2. Therefore, ReWλ,μz>(1+α)/2.

(iv) To prove that Wλ,μ/zPα,  we have to show that gz-1<1,  where gz=Wλ,μz/z-α/1-α. By using the well-known triangle inequality(35)z1+z2z1+z2with the inequality Γμ+nΓμ+nλ, nN, and the inequality(36)n!μ+1n-1μ+1n-1,nN,we have(37)gz-1=11-αn=1Γμn!Γλn+μzn11-α1μn=11n!μ+1n-111-α1μn=11μ+1n-1=μ+1μ21-α.Therefore, Wλ,μ/zPα for 0<α<1-(μ+1)/μ2.

Putting α=0 in Theorem 6, we have the following result.

Corollary 7.

Let λ,μR and zU. Then the following assertions are true:

If λ1 and, μ>1+3, then Wλ,μS.

If λ1 and μ>2+22, then Wλ,μC.

If λ1 and μ>6+43, then Wλ,μK1/2.

If λ1 and μ>1+5/2, then Wλ,μ/zP.

Theorem 8.

If α0,1, μ0, and λ-1, then a sufficient condition for zWλ,μ to be in Tγα is (38)γWλ,μ11-α+1-αγ+2γ1-αWλ,μ1+Wλ,μ12.

Proof.

Consider the identity,(39)zWλ,μz=z+n=2bn-1zn.By using (9), we will only show that(40)n=2nγ-γ+1n-αbn-1λ,μ1-α,where bn-1(λ,μ)=Γμ/n!Γλn+μ.

Now,(41)n=2nγ-γ+1n-αbn-1λ,μ=γn=2n2bn-1λ,μ+1-γ1+αn=2nbn-1λ,μ+αγ-1n=2bn-1λ,μ.From (39), a little simplification yields(42)z2Wλ,μz+zWλ,μz=z+n=2nbn-1λ,μzn.Differentiating (42) two times with respect to z, we have(43)z3Wλ,μz+3z2Wλ,μz+zWλ,μz=z+n=2n2bn-1λ,μzn,(44)z4Wλ,μ+6z3Wλ,μz+7z2Wλ,μz+zWλ,μz=z+n=2n3bn-1λ,μzn.Now for z=1, the expressions (39), (42), and (43) become(45)Wλ,μ1=1+n=2bn-1,Wλ,μ1+Wλ,μ1=1+n=2nbn-1λ,μ,Wλ,μ1+3Wλ,μ1+Wλ,μ1=1+n=2n2bn-1λ,μ.By using the above expressions, (41) becomes(46)n=2nγ-γ+1n-αbn-1λ,μ=γWλ,μ1+1-αγ+2γWλ,μ1+1-αWλ,μ1-1,And it is bounded above by 1-α if (38) holds. Thus the proof is completed.

Corollary 9.

The normalized Wright function is starlike of order 0α<1 with respect to the origin if(47)Wλ,μ11-α+Wλ,μ12.

Theorem 10.

If α0,1, μ0, and λ-1, then a sufficient condition for zWλ,μ to be in Lγα is(48)γWλ,μ1+5γ+1-αγWλ,μ11-α+4γ-2αγ-α+31-αWλ,μ1+Wλ,μ12.

Proof.

Consider the identity(49)zWλ,μz=z+n=2bn-1zn.By using the (10), we will only show that(50)n=2nnγ-γ+1n-αbn-1λ,μ1-α,where bn-1(λ,μ)=Γμ/n!Γλn+μ.

Now,(51)n=2nnγ-γ+1n-αbn-1λ,μ=γn=2n3bn-1λ,μ+1-γ1+αn=2n2bn-1λ,μ+αγ-1n=2nbn-1λ,μ.Now for z=1 and using (42), (43), and (44), expression (51) becomes(52)n=2nnγ-γ+1n-αbn-1λ,μ=γWλ,μ1+7Wλ,μ1+6Wλ,μ1+Wλ,μ1-1+1-γ1+αWλ,μ1+3Wλ,μ1+Wλ,μ1-1+αγ-1Wλ,μ1+Wλ,μ1-1=γWλ,μ1+5γ+1-αγWλ,μ1+4γ-2αγ-α+3Wλ,μ1+1-αWλ,μ1-1and is bounded above by 1-α if (48) holds, which is the required result.

Corollary 11.

The normalized Wright function is convex of order α, 0α<1, with respect to the origin if(53)Wλ,μ11-α+3-α1-αWλ,μ1+Wλ,μ12.

3. Close-to-Convexity of Wright Functions with respect to Certain Functions

The work in this section is motivated by the works of Baricz, Orhan and Yagmur, and Ponnusamy and Vuorinen [13, 15, 22, 23]. In this section we will discuss some conditions on the parameters λ and μ under which the Wright functions are assured close-to-convex with respect to the functions(54)-log1-z,12log1+z1-z.By using Lemma 2, we will get the following results.

Theorem 12.

If μ1 and λ1/2, then zWλ,μ(z) is close-to-convex with respect to the function -log1-z.

Proof.

Set(55)fz=Wλ,μz=z+n=2bn-1zn.We have bn-1>0 for all n2 and b1Γμ/Γ1+μ1, by using the inequality Γλn+μΓn+μ. To prove that f(z) is close-to-convex with respect to the function -log1-z, we use Lemma 2. Therefore, we have to prove that nbn-1n2 is a decreasing sequence. By a short computation, we obtain(56)nbn-1-n+1bn=Γμn-1!nΓλn-1+μ-n+1nΓλn+μ,=Γμn-1!n2Γλn+μ-n+1Γλn-1+μnΓλn+μΓλn-1+μ>0.By using the conditions on parameters, we can easily observe that nbn-1-(n+1)bn>0 for all n2, and thus nbn-1n2 is a decreasing sequence. By Lemma 2, it follows that f(z) is close-to-convex with respect to the function -log1-z.

Theorem 13.

If μ1 and λ1, then zzWλ,μ(z2) is close-to-convex with respect to the function 1/2log(1+z)/(1-z).

Proof.

Set(57)fz=zWλ,μz2=z+n=2B2n-1z2n-1.Here B2n-1=bn-1=Γμ/(n-1)!Γλn-1+μ, and therefore we have b1=Γμ/Γλ+μ1 and B2n-1>0 for all n2. To prove our main result we will prove that (2n-1)bn-1n2 is a decreasing sequence. By a short computation, we obtain(58)2n-1bn-1-2n+1bn=Γμn-1!2n-1Γλn-1+μ-2n+1nΓλn+μ,=Γμn-1!2n2-nΓλn+μ-2n+1Γλn-1+μnΓλn+μΓλn-1+μ>0.By using the conditions on parameters, we can easily observe that (2n-1)bn-1-(2n+1)bn>0 for all n2, and thus (2n-1)bn-1n2 is a decreasing sequence. By Lemma 2 it follows that f(z) is close-to-convex with respect to the function 1/2log(1+z)/(1-z).

4. Hardy Spaces of Wright Functions

Hardy spaces of hypergeometric functions are recently studied by Ponnusamy . Baricz  used the idea of Ponnusamy and found the Hardy spaces of Bessel functions. Yagmur and Orhan  studied the same problem for generalized Struve functions. Similarly, Yagmur  studied the problem for Lommel functions. For Hardy spaces related to some classes of analytic functions, we refer to .

Theorem 14.

Let, α0,1, μ>4-α+5α2-28α+32/21-α. Then

Wλ,μH1/1-2α    for α0,1/2.

Wλ,μH    for α1/2.

Proof.

From the definition of Hypergeometric function, we have(59)F21a,b,c;z=n=0anbncnznn!.Now, we have(60)k+dz1-zeiγ1-2α=k+dzF211,1-2α,1;zeiγ=k+dn=01-2αnn!eiγnzn+1,where k,dC, α1/2, and γ is any real number. Also, we have(61)k+dlog1-zeiγ=k-dzF211,1,z;zeiγ=k-dn=01n+1eiγnzn+1.This implies that Wλ,μ is not of the forms k+dz1-ziγ2α-1 for α1/2 and k+dlog1-zeiγ for α=1/2    , respectively.    Also from Theorem 6(ii) Wλ,μ is convex of order α. Hence, by using Lemma 5, we have the required result.

Theorem 15.

Let λ1, μ>1+5/2, and fR, and then the convolution Wλ,μf is in HR.

Proof.

Let hz=Wλ,μzfz. Then it is clear that hz=Wλ,μz/zfz. Using Corollary 7(iv), we have Wλ,μz/zP. Since fR, therefore by using Lemma 3, we have hR. It is also clear that Wλ,μz/z is an entire function and therefore h is entire. This implies that h is bounded. Hence, we have the required result.

Theorem 16.

Let λ,μR with λ1, μ>1+5-4α/21-α, α0,1, and zU. If fRβ, then Wλ,μfRγ, where γ=1-21-α1-β.

Proof.

Let hz=Wλ,μzfz. Then, hz=Wλ,μz/zfz. Now from Theorem 6(iv), we have Wλ,μz/zPα. By using Lemma 4 and the fact that fPβ, we have hzPγ, where γ=1-21-α1-β. Consequently, we have hRγ.

Corollary 17.

Let 0α<1, λ1, and μ>1+5-4α/21-α. If fRβ, β=1-2α2-2α, then Wλ,μzfzR0.

Corollary 18.

Let λ1 and μ>1+5/2. If fR1/2, then Wλ,μzfzR0.

5. Particular Case

At the end of this paper, we give some particular cases of the above-mentioned theorem. When we put λ=1 and μ=5/2 in (16), we obtain the function(62)W1,5/2-z=34sin2z2z-cos2z.By using Theorem 6 assertions (i) and (iv), we get the following corollary.

Corollary 19.

(i) If 0α<α0, where α00.0332, then W1,5/2Sα.

(ii) If 0α<α1, where α10.4310, then W1,5/2/zP(α).

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of Sarfraz Nawaz Malik is supported by COMSATS Institute of Information Technology, Pakistan, Reference no. 3-64/IPF-SRG/CIIT/Wah/14/765.

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