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 Technology3-64/IPF-SRG/CIIT/Wah/14/7651. 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∞=supz∈Ufz.We denote Hp, 0<p<∞, for the space of all functions f∈H 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 z∈U. Another equivalent definition of norm is given as follows: let f∈H, and set(2)Mpr,f=12π∫02πfreiθpdθ1/p,0<p<∞,maxfz:z≤r,p=∞.Then the function f∈Hp if Mpr,f is bounded for all r∈0,1. It is clear that (3)H∞⊂Hq⊂Hp,0<q<p<∞.For some details, see [1]. It is also known [1] that Ref′z>0 in U, and then (4)f′∈Hq,q<1,f∈Hq/1-q,0<q<1.Let A be the class of functions f of the form (5)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, and Kα denote the classes of starlike, convex, and close-to-convex functions of order α, respectively, and they are defined as(6)S∗α=f:f∈A,Rezf′zfz>α,z∈U,α∈0,1,Cα=f:f∈A,Re1+zf′′zf′z>α,z∈U,α∈0,1,Kα=f:f∈A,Rezf′zgz>α,z∈U,α∈0,1,g∈S∗.It is clear that (7)S∗0=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=f∈A:Rezf′z+γz2f′′z1-γfz+γzf′z>α,Lγα=f∈A:Reγz3f′′′z+1+2γz2f′′z+zf′zzf′z+γz2f′′z>α,where 0≤α,γ<1. The purpose of these subclasses is that when we put γ=0 in (8), we get T0α=S∗α and L0α=Cα. The sufficient coefficient conditions by which a function f∈A as defined in (5) belongs to the classes Tγα and Lγα are(9)∑n=2∞nγ-γ+1n-αan≤1-α,(10)∑n=2∞nnγ-γ+1n-αan≤1-α,respectively. For some details about these classes, see [2, 3]. Recently, Baricz [4] introduced the classes(11)Pηα=p:p∈H,p0=1,Reeiηpz-α>0,z∈U,α∈0,1,η∈R,Rηα=f:f∈A,Reeiηf′z-α>0,z∈U,α∈0,1,η∈R.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 f∈A given by (5) and g∈A given by(12)gz=z+∑n=2∞bnzn,and then Hadamard product (or convolution) of f and g is defined as(13)f∗gz=z+∑n=2∞anbnznz∈U.Recently, Prajapat [5] studied some geometric properties of Wright function(14)Wλ,μz=∑n=0∞znn!Γλn+μ,λ>-1,μ∈C.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 [6] 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 [9]. Prajapat [5] discussed some geometric properties of the Wright functions,(16)Wλ,μz=∑n=0∞Γμn!Γλn+μzn,λ>-1,μ>0,z∈U,and their normalization of the form(17)zWλ,μz=Wλ,μz=z+∑n=1∞Γμn!Γλn+μzn+1,λ>-1,μ>0,z∈U,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, 12–14], 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>]).

Iff∈A satisfies the inequality(18)zf′′z<1-α4,z∈U,0≤α<1,then(19)Ref′z>1+α4,z∈U,0≤α<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 1≥2a2≥⋯≥nan≥⋯≥0 or 1≤2a2≤⋯≤nan≤⋯≤2, 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 1≥3b3≥⋯≥(2n+1)b2n+1≥⋯≥0 or 1≤3b3≤⋯≤(2n+1)b2n+1≤⋯≤2, then g(z) is univalent in U.

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

P0α∗P0β⊂P0γ, 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<1and γ=1-21-α1-β, we have R0α∗R0β⊂R0γ or equivalently Pα∗P0β⊂P0γ.

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

If the function f, convex of order α, where α∈0,1, is not of the form(20)fz=k+dz1-zeiγ2α-1,α≠12,k+dlog1-zeiγ,α=12for some complex numbers k and d and for some real number γ, then the following statements hold:

There exists δ=δf>0 such that f′∈Hδ+1/21-α.

If α∈0,1/2, then there exists τ=τf>0 such that f∈Hτ+1/1-2α.

If α≥1/2, then f∈H∞.

2. Main ResultsTheorem 6.

Let λ,μ∈R with α∈0,1 and z∈U. Then the following assertions are true:

If λ≥1 and μ>2-α+5α2-16α+12/21-α, then Wλ,μ∈S∗α.

If λ≥1 and μ>4-α+5α2-28α+32/21-α, then Wλ,μ∈Cα.

If λ≥1 and μ>6+48-12α/1-α, then Wλ,μ∈K(1+α)/2.

If λ≥1 and μ>1+5-4α/21-α, then Wλ,μ/z∈Pα.

Proof.

(i) To prove that Wλ,μ∈S∗α,we have to show that zWλ,μ′(z)/Wλ,μ(z)-1<1-α. By using the well-known triangle inequality(21)z1+z2≤z1+z2with the inequality Γμ+n≤Γμ+nλ, n∈N, which is equivalent to Γμ/Γλn+μ≤1/μμ+1⋯(μ+n-1)=1/μn, n∈N, and the inequality(22)n!μ+1n-1≥nμ+1n-1,n∈N,we obtain(23)Wλ,μ′z-Wλ,μzz≤∑n=1∞nΓμn!Γλn+μ≤∑n=1∞nΓμn!Γn+μλ≥1≤1μ∑n=1∞1μ+1n-1=μ+1μ2,μ>0,z∈U.Also consider (24)Wλ,μzz=1+∑n=1∞Γμn!Γλn+μzn.Since Γn+μ≤Γλn+μ, therefore by using the reverse triangle inequality (25)z1-z2≤z1+z2and the inequality, n!μ+1n-1≥μ+1n-1, n∈N, we get(26)Wλ,μzz=1+∑n=1∞Γμn!Γλn+μzn≥1-∑n=1∞1n!μn≥1-1μ∑n=1∞1μ+1n-1=μ2-μ-1μ2,μ>0.By combining (23) and (26), we get(27)zWλ,μ′zWλ,μz-1≤μ+1μ2-μ-1<1-α.So Wλ,μ(z) is starlike function of order α, where 0≤α<1-(μ+1)/(μ2-μ-1).

(ii) To prove that Wλ,μ∈Cα,we have to show that zWλ,μ′′(z)/Wλ,μ′(z)<1-α. By using the well-known triangle inequality(28)z1+z2≤z1+z2with the inequality Γμ+n≤Γμ+nλ, n∈N, which is equivalent to Γμ/Γλn+μ≤1/μμ+1⋯μ+n-1=1/μn, n∈N, and the inequalities(29)n!≥nn+12n,μ+1n-1≥μ+1n-1,n∈N,we have (30)zWλ,μ′′z=∑n=1∞Γμnn+1n!Γλn+μzn≤∑n=1∞nn+12nΓμnn+1Γλn+μ≤2μ∑n=1∞2μ+1n-1=2μ+1μμ-1,μ>1.Since Γn+μ≤Γλn+μ, therefore by using the reverse triangle inequality (31)z1-z2≤z1+z2and the inequality, n!μ+1n-1≥nμ+1n-1, n∈N, we get(32)Wλ,μ′z≥1-∑n=1∞nΓμn!Γn+μ=1-1μ∑n=1∞nn!μ+1n-1≥1-1μ∑n=1∞1μ+1n-1=μ2-μ-1μ2,μ>0.Combining (30) and (32), we have(33)zWλ,μ′′zWλ,μ′z≤3μ+1μ2-μ-1<1-α.This implies that Wλ,μ is convex function of order α, where 0≤α<1-3μ+1/μ2-μ-1.

(iii) Using inequality (30) and Lemma 1, we have(34)zWλ,μ′′z≤3μ+1μ2≤1-α4,where 0≤α<1-12μ+1/μ2 and μ>6+48-12α/1-α. This shows that Wλ,μ∈K(1+α)/2. Therefore, ReWλ,μ′z>(1+α)/2.

(iv) To prove that Wλ,μ/z∈Pα,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+z2≤z1+z2with the inequality Γμ+n≤Γμ+nλ, n∈N, and the inequality(36)n!μ+1n-1≥μ+1n-1,n∈N,we have(37)gz-1=11-α∑n=1∞Γμn!Γλn+μzn≤11-α1μ∑n=1∞1n!μ+1n-1≤11-α1μ∑n=1∞1μ+1n-1=μ+1μ21-α.Therefore, Wλ,μ/z∈Pα for 0<α<1-(μ+1)/μ2.

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

Corollary 7.

Let λ,μ∈R and z∈U. Then the following assertions are true:

If λ≥1 and, μ>1+3, then Wλ,μ∈S∗.

If λ≥1 and μ>2+22, then Wλ,μ∈C.

If λ≥1 and μ>6+43, then Wλ,μ∈K1/2.

If λ≥1 and μ>1+5/2, then Wλ,μ/z∈P.

Theorem 8.

If α∈0,1, μ≥0, and λ≥-1, then a sufficient condition for zWλ,μ to be in Tγα is (38)γWλ,μ′′11-α+1-αγ+2γ1-αWλ,μ′1+Wλ,μ1≤2.

Proof.

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

Now,(41)∑n=2∞nγ-γ+1n-αbn-1λ,μ=γ∑n=2∞n2bn-1λ,μ+1-γ1+α∑n=2∞nbn-1λ,μ+αγ-1∑n=2∞bn-1λ,μ.From (39), a little simplification yields(42)z2Wλ,μ′z+zWλ,μz=z+∑n=2∞nbn-1λ,μzn.Differentiating (42) two times with respect to z, we have(43)z3Wλ,μ′′z+3z2Wλ,μ′z+zWλ,μz=z+∑n=2∞n2bn-1λ,μzn,(44)z4Wλ,μ′′′′+6z3Wλ,μ′′z+7z2Wλ,μ′z+zWλ,μz=z+∑n=2∞n3bn-1λ,μzn.Now for z=1, the expressions (39), (42), and (43) become(45)Wλ,μ1=1+∑n=2∞bn-1,Wλ,μ′1+Wλ,μ1=1+∑n=2∞nbn-1λ,μ,Wλ,μ′′1+3Wλ,μ′1+Wλ,μ1=1+∑n=2∞n2bn-1λ,μ.By using the above expressions, (41) becomes(46)∑n=2∞nγ-γ+1n-αbn-1λ,μ=γWλ,μ′′1+1-αγ+2γWλ,μ′1+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 0≤α<1 with respect to the origin if(47)Wλ,μ′11-α+Wλ,μ1≤2.

Theorem 10.

If α∈0,1, μ≥0, and λ≥-1, then a sufficient condition for zWλ,μ to be in Lγα is(48)γWλ,μ′′′1+5γ+1-αγWλ,μ′′11-α+4γ-2αγ-α+31-αWλ,μ′1+Wλ,μ1≤2.

Proof.

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

Now,(51)∑n=2∞nnγ-γ+1n-αbn-1λ,μ=γ∑n=2∞n3bn-1λ,μ+1-γ1+α∑n=2∞n2bn-1λ,μ+αγ-1∑n=2∞nbn-1λ,μ.Now for z=1 and using (42), (43), and (44), expression (51) becomes(52)∑n=2∞nnγ-γ+1n-αbn-1λ,μ=γWλ,μ′′′1+7Wλ,μ′′1+6Wλ,μ′1+Wλ,μ1-1+1-γ1+αWλ,μ′′1+3Wλ,μ′1+Wλ,μ1-1+αγ-1Wλ,μ′1+Wλ,μ1-1=γWλ,μ′′′1+5γ+1-αγWλ,μ′′1+4γ-2αγ-α+3Wλ,μ′1+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, 0≤α<1, with respect to the origin if(53)Wλ,μ′′11-α+3-α1-αWλ,μ′1+Wλ,μ1≤2.

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 μ≥1 and λ≥1/2, then z→Wλ,μ(z) is close-to-convex with respect to the function -log1-z.

Proof.

Set(55)fz=Wλ,μz=z+∑n=2∞bn-1zn.We have bn-1>0 for all n≥2 and b1≤Γμ/Γ1+μ≤1, by using the inequality Γλn+μ≥Γn+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-1n≥2 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 n≥2, and thus nbn-1n≥2 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 μ≥1 and λ≥1, then z→zWλ,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=2∞B2n-1z2n-1.Here B2n-1=bn-1=Γμ/(n-1)!Γλn-1+μ, and therefore we have b1=Γμ/Γλ+μ≤1 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. 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 n≥2, and thus (2n-1)bn-1n≥2 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 [24]. Baricz [4] used the idea of Ponnusamy and found the Hardy spaces of Bessel functions. Yagmur and Orhan [25] studied the same problem for generalized Struve functions. Similarly, Yagmur [26] studied the problem for Lommel functions. For Hardy spaces related to some classes of analytic functions, we refer to [27–30].

Theorem 14.

Let, α∈0,1, μ>4-α+5α2-28α+32/21-α. Then

Wλ,μ∈H1/1-2αfor α∈0,1/2.

Wλ,μ∈H∞for α≥1/2.

Proof.

From the definition of Hypergeometric function, we have(59)F21a,b,c;z=∑n=0∞anbncnznn!.Now, we have(60)k+dz1-zeiγ1-2α=k+dzF211,1-2α,1;zeiγ=k+d∑n=0∞1-2αnn!eiγnzn+1,where k,d∈C, α≠1/2, and γ is any real number. Also, we have(61)k+dlog1-zeiγ=k-dzF211,1,z;zeiγ=k-d∑n=0∞1n+1eiγnzn+1.This implies that Wλ,μ is not of the forms k+dz1-ziγ2α-1 for α≠1/2 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 λ≥1, μ>1+5/2, and f∈R, and then the convolution Wλ,μ∗f is in H∞∩R.

Proof.

Let hz=Wλ,μz∗fz. Then it is clear that h′z=Wλ,μz/z∗f′z. Using Corollary 7(iv), we have Wλ,μz/z∈P. Since f∈R, therefore by using Lemma 3, we have h∈R. 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,μ∈R with λ≥1, μ>1+5-4α/21-α, α∈0,1, and z∈U. If f∈Rβ, then Wλ,μ∗f∈Rγ, where γ=1-21-α1-β.

Proof.

Let hz=Wλ,μz∗fz. Then, h′z=Wλ,μz/z∗f′z. Now from Theorem 6(iv), we have Wλ,μz/z∈Pα. By using Lemma 4 and the fact that f′∈Pβ, we have h′z∈Pγ, where γ=1-21-α1-β. Consequently, we have h∈Rγ.

Corollary 17.

Let 0≤α<1, λ≥1, and μ>1+5-4α/21-α. If f∈Rβ, β=1-2α2-2α, then Wλ,μz∗fz∈R0.

Corollary 18.

Let λ≥1 and μ>1+5/2. If f∈R1/2, then Wλ,μz∗fz∈R0.

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 0≤α<α0, where α0≃0.0332⋯, then W1,5/2∈S∗α.

(ii) If 0≤α<α1, where α1≃0.4310⋯, then W1,5/2/z∈P(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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