AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 954513 10.1155/2013/954513 954513 Research Article Starlikeness and Convexity of Generalized Struve Functions 0000-0001-5248-982X Yagmur Nihat 1 Orhan Halit 2 Bayram Mustafa 1 Department of Mathematics Faculty of Science and Art Erzincan University 24000 Erzincan Turkey erzincan.edu.tr 2 Department of Mathematics Faculty of Science Ataturk University 25240 Erzurum Turkey atauni.edu.tr 2013 6 3 2013 2013 03 12 2012 14 01 2013 2013 Copyright © 2013 Nihat Yagmur and Halit Orhan. 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.

We give sufficient conditions for the parameters of the normalized form of the generalized Struve functions to be convex and starlike in the open unit disk.

1. Introduction and Preliminary Results

It is well known that the special functions (series) play an important role in geometric function theory, especially in the solution by de Branges of the famous Bieberbach conjecture. The surprising use of special functions (hypergeometric functions) has prompted renewed interest in function theory in the last few decades. There is an extensive literature dealing with geometric properties of different types of special functions, especially for the generalized, Gaussian, and Kummer hypergeometric functions and the Bessel functions. Many authors have determined sufficient conditions on the parameters of these functions for belonging to a certain class of univalent functions, such as convex, starlike, and close-to-convex functions. More information about geometric properties of special functions can be found in . In the present investigation our goal is to determine conditions of starlikeness and convexity of the generalized Struve functions. In order to achieve our goal in this section, we recall some basic facts and preliminary results.

Let 𝒜 denote the class of functions f normalized by (1)f(z)=z+n2anzn, which are analytic in the open unit disk 𝒰={z:|z|<1}. Let 𝒮 denote the subclass of 𝒜 which are univalent in 𝒰. Also let 𝒮*(α) and 𝒞(α) denote the subclasses of 𝒜 consisting of functions which are, respectively, starlike and convex of order α in 𝒰(0α<1). Thus, we have (see, for details, ), (2)𝒮*(α)={f:f𝒜and(zf(z)f(z))>α,(z𝒰;  0α<1){(zf(z)f(z))}},𝒞(α)={f:f𝒜and(1+zf′′(z)f(z))>α,(z𝒰;  0α<1){(zf(z)f(z))}}, where, for convenience, (3)𝒮*(0)=𝒮*,𝒞(0)=𝒞. We remark that, according to the Alexander duality theorem , the function f:𝒰 is convex of order α, where 0α<1 if and only if zzf(z) is starlike of order α. We note that every starlike (and hence convex) function of the form (1) is univalent. For more details we refer to the papers in [10, 12, 13] and the references therein.

Denote by 𝒮1*(α), where α[0,1), the subclass of 𝒮*(α) consisting of functions f for which (4)|zf(z)f(z)-1|<1-α, for all z𝒰. A function f is said to be in 𝒞1(α) if zf𝒮1*(α).

Lemma 1 (see [<xref ref-type="bibr" rid="B12">4</xref>]).

If f𝒜 and (5)|zf(z)f(z)-1|1-β|zf′′(z)f(z)|β<(1-α)1-2β(1-3α2+α2)β, for some fixed α[0,1/2] and β0, and for all z𝒰, then f is in the class 𝒮*(α).

Lemma 2 (see [<xref ref-type="bibr" rid="B16">14</xref>]).

Let α[0,1). A sufficient condition for f(z)=z+n2anzn to be in 𝒮1*(α) and 𝒞1(α), respectively, is that (6)n2(n-α)|an|1-α,n2n(n-α)|an|1-α, respectively.

Lemma 3 (see [<xref ref-type="bibr" rid="B16">14</xref>]).

Let α[0,1). Suppose that f(z)=z-n2anzn,an0. Then a necessary and sufficient condition for f to be in 𝒮1*(α) and 𝒞1(α), respectively, is that (7)n2(n-α)|an|1-α,n2n(n-α)|an|1-α, respectively. In addition f𝒮1*(α)f𝒮*(α),  f𝒞1(α)f𝒞(α), and f𝒮*f𝒮.

2. Starlikeness and Convexity of Generalized Struve Functions

Let us consider the second-order inhomogeneous differential equation [15, page 341] (8)z2w′′(z)+zw(z)+(z2-p2)w(z)=4(z/2)p+1πΓ(p+1/2) whose homogeneous part is Bessel’s equation, where p is an unrestricted real (or complex) number. The function Hp, which is called the Struve function of order p, is defined as a particular solution of (8). This function has the form (9)Hp(z)=n0(-1)nΓ(n+3/2)Γ(p+n+3/2)(z2)2n+p+1,z. The differential equation (10)z2w′′(z)+zw(z)-(z2+p2)w(z)=4(z/2)p+1πΓ(p+1/2), which differs from (8) only in the coefficient of w. The particular solution of (10) is called the modified Struve function of order p and is defined by the formula [15, page 353] (11)Lp(z)=-ie-ipπ/2Hp(iz)=n01Γ(n+3/2)Γ(p+n+3/2)(z2)2n+p+1,z. Now, let us consider the second-order inhomogeneous linear differential equation , (12)z2w′′(z)+bzw(z)+[cz2-p2+(1-b)p]w(z)=4(z/2)p+1πΓ(p+b/2), where b,c,p. If we choose b=1and c=1, then we get (8), and if we choose b=1andc=-1, then we get (10). So this generalizes (8) and (10). Moreover, this permits to study the Struve and modified Struve functions together. A particular solution of the differential equation (12), which is denoted by wp,b,c(z), is called the generalized Struve function  of order p. In fact we have the following series representation for the function wp,b,c(z): (13)wp,b,c(z)=n0(-1)ncnΓ(n+3/2)Γ(p+n+(b+2)/2)(z2)2n+p+1,z. Although the series defined in (13) is convergent everywhere, the function wp,b,c(z) is generally not univalent in 𝒰. Now, consider the function up,b,c(z) defined by the transformation (14)up,b,c(z)=2pπΓ(p+b+22)z(-p-1)/2wp,b,c(z). By using the Pochhammer (or Appell) symbol, defined in terms of Euler’s gamma functions, by (λ)n=Γ(λ+n)/Γ(λ)=λ(λ+1)(λ+n-1), we obtain for the function up,b,c(z) the following form: (15)up,b,c(z)=n0(-c/4)n(3/2)n(κ)nzn=b0+b1z+b2z2++bnzn+, where κ=p+(b+2)/20,-1,-2,. This function is analytic on and satisfies the second-order inhomogeneous differential equation (16)4z2u′′(z)+2(2p+b+3)zu(z)+(cz+2p+b)u(z)=2p+b. Orhan and Yaǧmur  have determined various sufficient conditions for the parameters p,b, and c such that the functions up,b,c(z) or zzup,b,c(z) to be univalent, starlike, convex, and close to convex in the open unit disk. In this section, our aim is to complete the above-mentioned results.

For convenience, we use the notations: wp,b,c(z)=wp(z) and up,b,c(z)=up(z).

Proposition 4 (see [<xref ref-type="bibr" rid="B11">16</xref>]).

If b,c,p, κ=p+(b+2)/21,0,-1,-2,, and z, then for the generalized Struve function of order p the following recursive relations hold:

zwp-1(z)+czwp+1(z)=(2κ-3)wp(z)+2(z/2)p+1/πΓ(κ);

zwp(z)+(p+b-1)wp(z)=zwp-1(z);

zwp(z)+czwp+1(z)=pwp(z)+2(z/2)p+1/πΓ(κ);

[z-pwp(z)]=-cz-pwp+1(z)+1/2pπΓ(κ);

up(z)+2zup(z)+(cz/2κ)up+1(z)=1.

Theorem 5.

If the function up, defined by (15), satisfies the condition (17)|zup(z)up(z)|<1-α, where α[0,1/2] and z𝒰, then zup𝒮*(α).

Proof.

If we define the function g:𝒰 by g(z)=zup(z) for z𝒰. The given condition becomes (18)|zg(z)g(z)-1|<1-α, where z𝒰. By taking β=0 in Lemma 1, we thus conclude from the previous inequality that g𝒮*(α), which proves Theorem 5.

Theorem 6.

If the function up, defined by (15), satisfies the condition (19)|zup′′(z)up(z)|<1-3α/2+α21-α, where α[0,1/2] and z𝒰, then it is starlike of order α with respect to 1.

Proof.

Define the function h:𝒰 by h(z)=[up(z)-b0]/b1. Then h𝒜 and (20)|zh′′(z)h(z)|=|zup′′(z)up(z)|<1-3α/2+α21-α, where α[0,1/2] and z𝒰. By taking β=1 in Lemma 1, we deduce that h𝒮*(α); that is, h is starlike of order α with respect to the origin for α[0,1/2]. So, Theorem 6 follows from the definition of the function h, because b0=1.

Theorem 7.

If for α[0,1/2] and c0 one has (21)|zup+1(z)up+1(z)|<1-α, for all z𝒰, then up+2zup is starlike of order α with respect to 1.

Proof.

Theorem 5 implies that zup+1𝒮*(α). On the other hand, the part (v) of Proposition 4 yields (22)up(z)+2zup(z)=-c2κzup+1(z)+1. Since the addition of any constant and the multiplication by a nonzero quantity do not disturb the starlikeness. This completes the proof.

Lemma 8.

If b,p, c, and κ=p+(b+2)/2 such that κ>|c|/2, then the function up:𝒰 satisfies the following inequalities: (23)6κ-2|c|6κ-|c||up(z)|6κ6κ-|c|,(24)|c|(2κ-|c|)3κ(4κ-|c|)|up(z)|2|c|3(4κ-|c|),(25)|zup′′(z)||c|24κ(4κ-|c|).

Proof.

We first prove the assertion (23) of Lemma 8. Indeed, by using the well-known triangle inequality: (26)|z1+z2||z1|+|z2|, and the inequalities (3/2)n(3/2)n, (κ)nκn(n), we have (27)|up(z)|=|1+n1(-c/4)n(3/2)n(κ)nzn|1+n1(|-c/4|(3/2)κ)n=1+|c|6κn1(|c|6κ)n-1=6κ6κ-|c|,(κ>|c|6). Similarly, by using reverse triangle inequality: (28)|z1-z2|||z1|-|z2||, and the inequalities (3/2)n(3/2)n, (κ)nκn    (n), then we get (29)  |up(z)|=|1+n1(-c/4)n(3/2)n(κ)nzn|1-n1(|-c/4|(3/2)κ)n=1-|c|6κn1(|c|6κ)n-1=6κ-2|c|6κ-|c|,(κ>|c|6), which is positive if κ>|c|/3.

In order to prove assertion (24) of Lemma 8, we make use of the well-known triangle inequality and the inequalities (3/2)n(3/2)n, (κ)nκn(n), and we obtain (30)|up(z)|=|n1n(-c/4)n(3/2)n(κ)nzn-1|23n1(|c|4κ)n=23|c|4κn1(|c|4κ)n-1=2|c|3(4κ-|c|),(κ>|c|4). Similarly, by using the reverse triangle inequality and the inequalities (3/2)n(3/2)n, (κ)nκn(n), we have (31)|up(z)|=|n1n(-c/4)n(3/2)n(κ)nzn-1|  |c|6κ-23(|c|4κ)2n2(|c|4κ)n-1=|c|(2κ-|c|)3κ(4κ-|c|),(κ>|c|4), which is positive if κ>|c|/2.

We now prove assertion (25) of Lemma 8 by using again the triangle inequality and the inequalities (3/2)nn(n-1), (κ)nκn(n), and we arrive at the following: (32)|zup′′(z)|=|n2n(n-1)(-c/4)n(3/2)n(κ)nzn-1||c|4κn2(|c|4κ)n-1=|c|24κ(4κ-|c|),(κ>|c|4).

Thus, the proof of Lemma 8 is completed.

Theorem 9.

If b,p, c and κ=p+(b+2)/2, then the following assertions are true.

If κ>(7/8)|c|, then up(z) is convex in 𝒰.

If κ>((11+41)/24)|c|, then zup(z) is starlike of order 1/2 in 𝒰, and consequently the function zz-pwp(z) is starlike in 𝒰.

If κ>((11+41)/24)|c|-1, then the function zup(z)+2zup(z) is starlike of order 1/2 with respect to 1 for all z𝒰.

Proof.

(i) By combining the inequalities (24) with (25), we immediately see that (33)|zup′′(z)up(z)|3|c|4(2κ-|c|). So, for κ>(78)|c|, we have (34)|zup′′(z)up(z)|<1. This shows up(z) is convex in 𝒰.

(ii) If we let g(z)=zup(z) and h(z)=zup(z2), then (35)h(z)=g(z2)z=2pπΓ(κ)z-pwp,b,c(z),zh(z)h(z)-1=2[z2g(z2)g(z2)-1]=2z2up(z2)up(z2), so that (36)|zh(z)h(z)-1|<1,z𝒰, if and only if (37)|z2up(z2)up(z2)|<12,z𝒰.

It follows that zup(z) is starlike of order 1/2 if (37) holds.

From (24) and (23), we have (38)|z2up(z2)|2|c|3(4κ-|c|),(κ>|c|4),(39)6κ-2|c|6κ-|c||up(z2)|,(κ>|c|3), respectively.

By combining the inequalities (38) with (39), we see that (40)|z2up(z2)up(z2)||c|(6κ-|c|)3(3κ-|c|)(4κ-|c|), where κ>|c|/3, and the above bound is less than or equal to 1/2 if and only if κ>((11+41)/24)|c|. It follows that zup is starlike of order 1/2 in 𝒰 and z-pwp,b,c is starlike in 𝒰.

(iii) The part (ii) of Theorem 9 implies that for κ>((11+41)/24)|c|-1, the function zzup+1(z) is starlike of order 1/2 in 𝒰. On the other hand, the part (v) of Proposition 4 yields (41)up(z)+2zup(z)=-c2κzup+1(z)+1. So the function zup(z)+2zup(z) is starlike of order 1/2 with respect to 1 for all z𝒰.

This completes the proof.

Struve Functions. Choosing b=c=1, we obtain the differential equation (8) and the Struve function of order p, defined by (9), satisfies this equation. In particular, the results of Theorem 9 are as follows.

Corollary 10.

Let p:𝒰 be defined by p(z)=2pπΓ(p+3/2)z-p-1Hp(z)=up,1,1(z2), where Hp stands for the Struve function of order p. Then the following assertions are true.

If p>-5/8, then p(z1/2) is convex in 𝒰.

If p>(-25+41)/24, then zp(z1/2) is starlike of order 1/2 in 𝒰, and consequently the function zz-pHp(z) is starlike in 𝒰.

If p>(-49+41)/24, then the function zp(z1/2)+2zp(z1/2) is starlike of order 1/2 with respect to 1 for all z𝒰.

Modified Struve Functions. Choosing b=1  and  c=-1, we obtain the differential equation (10) and the modified Struve function of order p, defined by (11). For the function p:𝒰 defined by p(z)=2pπΓ(p+3/2)z-p-1Lp(z)=up,1,-1(z2), where Lp stands for the modified Struve function of order p. The properties are same like for function p, because we have |c|=1. More precisely, we have the following results.

Corollary 11.

The following assertions are true.

If p>-5/8, then p(z1/2) is convex in 𝒰.

If p>(-25+41)/24, then zp(z1/2) is starlike of order 1/2 in 𝒰, and consequently the function zz-pLp(z) is starlike in 𝒰.

If p>(-49+41)/24, then the function zp(z1/2)+2zp(z1/2) is starlike of order 1/2 with respect to 1 for all z𝒰.

Example 12.

If we take p=-1/2, then from part (ii) of Corollary 10, the function zz1/2H-1/2(z)=2/πsinz is starlike in 𝒰. So the function f(z)=sinz is also starlike in 𝒰. We have the image domain of f(z)=sinz illustrated by Figure 1.

f ( z ) = sin z .

Theorem 13.

If α[0,1),c<0, and κ>0, then a sufficient condition for zup to be in 𝒮1*(α) is (42)up(1)+up(1)1-α2. Moreover, (42) is necessary and sufficient for ψ(z)=z[2-up(z)] to be in 𝒮1*(α).

Proof.

Since zup(z)=z+n2bn-1zn, according to Lemma 2, we need only show that (43)n2(n-α)bn-11-α. We notice that (44)n2(n-α)bn-1=n2(n-1)bn-1+n2(1-α)bn-1=n2(n-1)(-c/4)n-1(3/2)n-1(κ)n-1+(1-α)[up(1)-1]=up(1)+(1-α)[up(1)-1]. This sum is bounded above by 1-α if and only if (42) holds. Since (45)z[2-up(z)]=z-n2bn-1zn, the necessity of (42) for ψ to be in 𝒮1*(α) follows from Lemma 3.

Corollary 14.

If c<0 and κ>0, then a sufficient condition for zup to be in 𝒮1*(1/2) is (46)up+1(1)-2κc. Moreover, (46) is necessary and sufficient for ψ(z)=z[2-up(z)] to be in 𝒮1*(1/2).

Proof.

For α=1/2, the condition (42) becomes up(1)+2up(1)2. From the part (v) of Proposition 4 we get (47)up(1)+2zup(1)=1-c2κup+1(1). So, up(1)+2up(1)2 if and only if 1-(c/2κ)up+1(1)2. Thus, we obtain the condition (46).

Furthermore, from the proof of Theorem 13, we have necessary and sufficient condition for ψ(z)=z[2-up(z)] to be in 𝒮1*(1/2).

Theorem 15.

If α[0,1),c<0 and κ>0, then a sufficient condition for zup to be in 𝒞1(α) is (48)up′′(1)+(3-α)up(1)+(1-α)up(1)-2α2. Moreover, (48) is necessary and sufficient for ψ(z)=z[2-up(z)] to be in 𝒞1(α).

Proof.

In view of Lemma 2, we need only to show that (49)n2n(n-α)bn-11-α. If we let g(z)=zup(z), we notice that (50)  n2n(n-α)bn-1=n2n(n-1)bn-1+(1-α)n2nbn-1=g′′(1)+(1-α)[g(1)-1]=up′′(1)+(3-α)up(1)+(1-α)up(1)-1+α. This sum is bounded above by 1-α if and only if (48) holds. Lemma 3 implies that (48) is also necessary for ψ to be in 𝒞1(α).

Theorem 16.

If c<0,κ>0, and up(1)2, then 0zup(t)dt𝒮*.

Proof.

Since (51)0zup(t)dt=n0bnn+1zn+1=z+n2bn-1nzn, we note that (52)n2nbn-1n=n2bn-1=up(1)-11, if and only if up(1)2.

Acknowledgment

The present paper was supported by Ataturk University Rectorship under The Scientific and Research Project of Ataturk University, Project no: 2012/173.

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