JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 784560 10.1155/2013/784560 784560 Research Article Certain Properties of Multivalent Functions Associated with the Dziok-Srivastava Operator Seoudy T. M. Benson Harold Department of Mathematics Faculty of Science, Fayoum 63514 Egypt fayoum.edu.eg 2013 22 1 2013 2013 11 10 2012 26 10 2012 2013 Copyright © 2013 T. M. Seoudy. 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.

By making use of the techniques of the differential subordination, we derive certain properties of p-valent functions associated with the Dziok-Srivastava operator.

1. Introduction

Let A(p,k) denote the class of functions of the form (1)f(z)=zp+n=kan+pzn+p(p,k={1,2,3,}), which are analytic in the open unit disk  U={z:|z|<1}. We write A(p,1)=A(p).

Suppose that f and g are analytic in U. We say that the function f is subordinate to g in U, or g superordinate tof in U, and we write fg or f(z)g(z)(zU), if there exists an analytic function ω in U with ω(0)=0 and |ω(z)|<1, such that f(z)=g(ω(z))(zU). If g is univalent in U, then the following equivalence relationship holds true (see ): (2)f(z)g(z)f(0)=g(0),f(U)g(U).

For functions fjA(p,k) given by (3)fj(z)=zp+n=kan+p,jzn+p(j=1,2;p), we define the Hadamard product (or convolution) of f1 and f2 by (4)(f1*f2)(z)=zp+n=kan+p,1an+p,2zn+p=(f2*f1)(z).

For complex parameters a1,,aq and b1,,bs(bj0-={0,-1,-2,};j=1,,s), the generalized hypergeometric function Fsq is defined (see ) by the following infinite series: (5)  qFs(a1,,ai,,aq;b1,,bs;z)=n=0(a1)n(aq)n(b1)n(bs)nznn!(qs+1;q,s0={0};zU), where (θ)n is the Pochhammer symbol defined, in terms of the Gamma function Γ, by (6)(θ)n=Γ(θ+n)Γ(θ)={1,(υ=0),θ(θ+1)(θ+n-1),(υ). Corresponding a function hp(a1,,ai,,aq;b1,,bs;z) defined by (7)hp(a1,,ai,,aq;b1,,bs;z)=zp·qFs(a1,,ai,,aq;b1,,bs;z)(zU), Dziok and Srivastava  considered a linear operator (8)Hp(a1,,aq;b1,,bs):A(p,k)A(p,k) defined by the following Hadamard product: (9)Hp(a1,,aq;b1,,bs)f(z),=hp(a1,,ai,,aq;b1,,bs;z)*f(z),(qs+1;q,s0;zU). If fA(p,k) is given by (1), then we have (10)Hp(a1,,aq;b1,,bs)f(z)=zp+n=kΓnan+pzn+p(zU), where (11)Γn=(a1)n(aq)n(b1)n(bs)n1n!,(n). To make the notation simple, we write (12)Hp,q,s(a1)f(z)=Hp(a1,,aq;b1,,bs)f(z). It easily follows from (9) or (10) that (13)z(Hp,q,s(a1)f(z))=a1Hp,q,s(a1+1)f(z)-(a1-p)Hp,q,s(a1)f(z),(zU). It should be remarked that the linear operator Hp,q,s(a1) is a generalization of many other linear operators considered earlier. In particular, for fA(p) we have the following observations:

H1,2,1(a,b;c)f(z)=(Ica,b)f(z)(a,b;c0-), where the linear operator Ica,b was investigated by Hohlov ;

Hp,2,1(n+p,1;1)f(z)=Dn+p-1f(z)(n;n>-p), where the linear operator Dn+p-1 was studied by Goel and Sohi . In the case when p=1, Dnf(z) is the Ruscheweyh derivative of f(z) (see );

Hp,2,1(μ+p,1;μ+p+1)f(z)=Jp,δ(f)(z)=((p+δ)/zδ)0ztδ-1f(t)dt(δ>-p), where Jp,δ is the generalized Bernardi-Libera-Livingston integral operator (see );

Hp,2,1(p+1,1;p+1-λ)f(z)=Ωz(λ,p)f(z)=(Γ(p+1-λ)/Γ(p+1))zλDzλf(z)(-λ<p+1;zU), where Dzλf(z) is the fractional integral of f of order -λ when -λ<0 and fractional derivative of f of order λ when 0λ<p+1. The extended fractional differintegral operator Dz(λ,p) was introduced and studied by Patel and Mishra . The fractional differential operator Ωz(λ,p) with 0λ<1 was investigated by Srivastava and Aouf . The operator Ωz(λ,1)=Ωzλ was introduced by Owa and Srivastava  (see also ).

Hp,2,1(a,1;c)f(z) = Lp(a,c)f(z)  (a;c0-), where the linear operator Lp(a,c) was studied by Saitoh  which yields the operator L(a,c) introduced by Carlson and Shaffer  for p=1;

H1,2,1(μ,1;λ+1)f(z) = Iλ,μf(z)  (λ>-1;μ>0), where Iλ,μ is the Choi-Saigo-Srivastava operator  which is closely related to the Carlson-Shaffer  operator L(μ,λ+1)f(z);

Hp,2,1(p+1,1;n+p)f(z) = In,pf(z)  (n;n>-p), where the operator In,p was considered by Liu and Noor ;

Hp,2,1(λ+p,c;a)f(z) = Ipλ(a,c)f(z)  (a,c0-;λ>-p), where Ipλ(a,c) is the Cho-Kwon-Srivastava operator .

In recent years, many interesting subclasses of analytic functions, associated with the Dziok-Srivastava operator Hp,q,s(a1) and its many special cases, were investigated by, for example, Dziok and Srivastava [5, 20], Gangadharan et al. , Liu and Noor , Liu , Liu and Srivastava , and others (see also [19, 2426]). In the present paper, we shall use the method based upon the differential subordination to derive inclusion relationships and other interesting properties and characteristics of the Dziok-Srivastava operator Hp,q,s(a1).

2. Main Results

Unless otherwise mentioned, we assume throughout the sequel that ai>0; ai0-  (i=1,,q); α>0; μ>0 and -1B<A1.

Let P[k] denote the class of functions of the form (14)φ(z)=1+ckzk+ck+1zk+1+ that are analytic in U, we write P=P. In our present investigation, we shall require the following lemmas.

Lemma 1 (see [<xref ref-type="bibr" rid="B15">2</xref>]).

Let h be analytic and convex (univalent) in U with h(0)=1 and φP[k]. If (15)φ(z)+zφ(z)γh(z), then, for γ0 and (γ)0, (16)φ(z)q(z)=γkz-γ/k0ztγ/k-1h(t)dth(z), and q is the best dominant.

Lemma 2 (see [<xref ref-type="bibr" rid="B14">1</xref>]).

Let D be a set in the complex plane and b be a complex number satisfying (b)>0. Suppose that the function Ψ:2×U satisfies the condition Ψ(ix,y)D for all real x,y-|b-ix|/2(b) and for all zU. If the functions φP and {Ψ(φ(z),zφ(z);z)}D, then {φ(z)}>0 in U.

Lemma 3 (see [<xref ref-type="bibr" rid="B28">27</xref>]).

Let ϕ be analytic in U with ϕ(0)=1 and ϕ(z)0 for all zU. If there exist two points z1,z2U such that (17)-π2δ1=arg{ϕ(z1)}<arg{ϕ(z)}<arg{ϕ(z2)}=π2δ2 for some δ1 and δ2(δ1,δ2>0) and for all z(|z|<|z1|=|z2|), then (18)z1ϕ(z1)ϕ(z1)=-i(δ1+δ22m),z2ϕ(z2)ϕ(z2)=-i(δ1+δ22m), where (19)m1-|b|1+|b|,b=itan(δ2-δ1δ2+δ1).

Theorem 4.

Let m1, γ>0. Let fA(k,p), then (20){(Hp,q,s(a1+1)f(z))(j)(Hp,q,s(a1)f(z))(j)}<a1+γa1(zU;0j<p), implies (21){((Hp,q,s(a1+1)f(z))(j)zp-j)-1/2γm}>2-1/m(zU;0j<p). The bound 2-1/m is the best possible.

Proof.

It easily follows from (13) that (22)z(Hp,q,s(a1)f(z))(j+1)=a1(Hp,q,s(a1+1)f(z))(j)-(a1-p+j)(Hp,q,s(a1)f(z))(j)(zU;0j<p). From (20) and (22), we have (23){z(Hp,q,s(a1+1)f(z))(j+1)(Hp,q,s(a1)f(z))(j)}<γ+p-j(zU;0j<p). That is, (24)-12γ(z(Hp,q,s(a1+1)f(z))(j+1)(Hp,q,s(a1)f(z))(j)-p+j)z1-z(zU). Let (25)φ(z)=((p-j)!p!(Hp,q,s(a1)f(z))(j)zp-j)-1/2γ(zU), then (24) may be written as (26)z(logφ(z))z(log11-z). By using a well-known result (see ) to (26) we obtain that (27)φ(z)11-z, or, equivalently, (28)((p-j)!p!(Hp,q,s(a1)f(z))(j)zp-j)-1/2γm=(11-ω(z))1/m, where ω is analytic in U, ω(0)=0 and |ω(z)|<1 for zU. Since (t1/m)((t))1/m for (t)>0 and m1, (28) yields (29)((p-j)!p!(Hp,q,s(a1)f(z))(j)zp-j)-1/2γm((11-ω(z)))1/m2-1/m(zU). To see that the bound 2-1/m cannot be increased, we consider the function (30)g(z)=zp+p!(p-j)!n=1(-2γ)n(n+p-j)!n!(n+p)!Γnzn+p,(zU). Since (31)(p-j)!p!(Hp,q,s(a1)g(z))(j)zp-j=(1-z)-2γ, we easily have that g satisfies (20) and (32)((p-j)!p!(Hp,q,s(a1)g(z))(j)zp-j)-1/2γm2-1/m as (z)=z1-. This completes the proof of Theorem 4.

Theorem 5.

Let α0, γ>1. If fA(p) satisfies the following inequality (33){(1-α)(Hp,q,s(a1+1)f(z))(j)(Hp,q,s(a1)f(z))(j)+α(Hp,q,s(a1+2)f(z))(j)(Hp,q,s(a1+1)f(z))(j)}<γ(0j<p;zU), then (34){(Hp,q,s(a1+1)f(z))(j)(Hp,q,s(a1)f(z))(j)}<β(0j<p;zU), where β(1,) is the positive root of the equation (35)2(a1-α+1)x2+(3α-2γα-2γ)x-α=0.

Proof.

Let (36)φ(z)=1β-1[β-(Hp,q,s(a1+1)f(z))(j)(Hp,q,s(a1)f(z))(j)](zU), then φ(z) is analytic in U and φ(0)=1. Differentiating (36) and using (22), we obtain that (37)(1-α)(Hp,q,s(a1+1)f(z))(j)(Hp,q,s(a1)f(z))(j)+α(Hp,q,s(a1+2)f(z))(j)(Hp,q,s(a1+1)f(z))(j)=β-α(β-1)a1+1-(a1-α+1)(β-1)a1+1φ(z)-α(β-1)a1+1zφ(z)β-(β-1)φ(z)=ψ(φ(z),zφ(z)), where (38)ψ(r,s)=β-α(β-1)a1+1-(a1-α+1)(β-1)a1+1r-α(β-1)a1+1sβ-(β-1)r. Using (33) and (38), we have (39){ψ(φ(z),zφ(z)):zU}D={z:(z)<γ}. Now for all real x,y-(1+x2)/2, we have (40){ψ(ix,y)}=β-α(β-1)a1+1-α(β-1)a1+1βyβ2+(β-1)2x2β-α(β-1)a1+1+αβ(β-1)2(a1+1)1+x2β2+(β-1)2x2β-α(β-1)a1+1+α(β-1)2β(a1+1)=β-α(β-1)(2β-1)2β(a1+1)=γ, where β is the positive root of (35).

Note that for α0, γ>1, a1>0 and (41)h(x)=2(a1-α+1)x2+(3α-2γα-2γ)x-α, we have h(0)=-α0 and h(1)=2a1(1-γ)-2γ<0. This shows β(0,+). Hence for each zU, ψ(ix,y)Ω. By Lemma 2, we get {φ(z)}>0(zU), and this proves (34).

Theorem 6.

Suppose that 0j<p; α>0 and 0<δ1, δ21. If Fα given by (42)Fα(z)=(1-α-αa1+αp)Hp,q,s(a1)f(z)+αa1Hp,q,s(a1+1)f(z) satisfies (43)-π2δ1<arg{Fα(j)(z)zp-j}<π2δ2(zU), then (44)-π2η1<arg{(Hp,q,s(a1)f(z))(j)zp-j}<π2η2(zU), where η1 and η2 are the solution of the equations: (45)δ1=η1+2πarctan[α(η1+η2)2(1-α+αp)(1-|b|1+|b|)],δ2=η2+2πarctan[α(η1+η2)2(1-α+αp)(1-|b|1+|b|)], where b is given by (19).

Proof.

Using (42) and the identity (22), it follows that (46)Fα(j)(z)=(1-α+αj)(Hp,q,s(a1)f(z))(j)+αz(Hp,q,s(a1)f(z))(j+1), for 0j<p. Putting (47)φ(z)=(p-j)!p!(Hp,q,s(a1)f(z))(j)zp-j(zU). On differentiating (47) followed by a simple calculation, we get (48)Fα(j)(z)zp-j=p!(1-α+αp)(p-j)!×{φ(z)+α1-α+αpzφ(z)}(zU). Let h be the function which maps U onto the angular domain {w:-(π/2)δ1<arg{w}<(π/2)δ2} with h(0)=1. By using (43) in (48), we get (49)φ(z)+α1-α+αpzφ(z)h(z). Further, an application of Lemma 1 yields {φ(z)}>0 in U and hence φ(z)0 for zU.

Suppose there exist two points z1,z2U such that the condition (28) is satisfied. Then by Lemma 3, we obtain (18) under the constraint (19). Therefore, we have (50)arg{(1-α+αp)φ(z1)+αzφ(z1)}=arg{φ(z1)}+arg{(1-α+αp)+αz1φ(z1)φ(z1)}=-π2η1+arg{(1-α+αp)-iα(η1+η2)2m}=-π2η1-arctan{α(η1+η2)2(1-α+αp)m}-π2η1-arctan{α(η1+η2)2(1-α+αp)(1-|b|1+|b|)},arg{(1-α+αp)φ(z2)+αzφ(z2)}-π2η2-arctan{α(η1+η2)2(1-α+αp)(1-|b|1+|b|)}, which contradicts the assumption (43). This proves the assertion (44) of the Theorem 6.

For δ1=δ2=δ, Theorem 6 reduces to the following corollary.

Corollary 7.

Suppose that 0j<p and α>0. If Fα defined by (42) satisfies (51)|arg{Fα(j)(z)zp-j}|<π2δ(0<δ1;zU), then (52)|arg{(Hp,q,s(a1)f(z))(j)zp-j}|<π2η(zU), where η(0<η1) is the solution of the equation: (53)δ=η+2πarctan(αη1-α+αp).

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