Paper29-Acta26-2011. 184KB Jun 04 2011 12:03:17 AM

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SANDWICH THEOREMS FOR ANALYTIC FUNCTIONS INVOLVING WRIGHT’S GENERALIZED HYPERGEOMETRIC

FUNCTION

M. K. Aouf, A. Shamandy, A. O. Mostafa, S. M. Madian

Abstract. In this paper, we obtain some applications of first order differential subordination and superordination results involving Wright’s generalized hypergeo-metric function defined by certain p-valent analytic functions.

2000Mathematics Subject Classification: 30C45. 1.Introduction Let A(p) denote the class of functions of the form:

f(z) =zp+ ∞

X

n=p+1

anzn (p∈N={1,2, ...}),1.1 (1)

which are analytic and p-valent in the open unit disk U ={z:z∈C,|z|<1}.

LetH(U) be the class of analytic functions inU and let H[a, p] be the subclass of

H(U) consisting of functions of the form:

f(z) =a+apzp+ap+1zp+1...(a∈C).

For simplicity, let H[a] = H[a,1]. Also, let A(1) = A be the subclass of H(U) consisting of functions of the form:

f(z) =z+a2z2+... .1.2 (2)

Iff, g∈H(U), we say thatf is subordinate to g, written f(z)≺g(z) if there exists a Schwarz function w(z), which (by definition) is analytic inU withw(0) = 0 and |w(z)|<1 for all z ∈U, such that f(z) =g(w(z)), z ∈U. Furthermore, if the functiong(z) is univalent in U,then we have the following equivalence, (cf., e.g.,[6], [19]; see also [20]):

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Letp, h∈H(U) and let ϕ(r, s, t;z) :C3×U →C.Ifp and ϕ(p(z), zp′ (z), z2_p′′

(z);z) are univalent and if p satisfies the second order superordination

h(z)≺ϕ(p(z), zp′

(z), z2p′′

(z);z),1.3 (3) thenpis a solution of the differential superordination (1.3).Note that iff is subordi-nate tog, thengis superordinate tof.An analytic functionqis called a subordinant ifq(z)≺p(z) for allpsatisfying (1.3). A univalent subordinantq_ethat satisfiesq ≺q_e

for all subordinants of (1.3) is called the best subordinant. Recently Miller and Mo-canu [21] obtained conditions on the functions h, q and ϕ for which the following implication holds:

h(z)≺ϕ(p(z), zp′

(z), z2p′′

(z);z)⇒q(z)≺p(z).1.4 (4) Using the results of Miller and Mocanu [21], Bulboaˇca [4] considered certain classes of first order differential superordinations as well as superordination-preserving inte-gral operators [5] many researshers ([1] and [29]) have obtained sufficient conditions on normalized analytic functions f(z) by means of differential subordinations and superordinations.

For analytic functions f(z) = ∞

P

n=0

anzn and g(z) = ∞

P

n=0

bnzn, the Hadamard product (orconvolution) of f(z) andg(z), is defined by

(f∗g) (z) = ∞

X

n=0

anbnzn.1.5 (5)

Letα1, A1, ..., αq, Aq and β1, B1, ..., βs, Bs (q, s∈N) be positive and real param-eters such that

1 + s

X

j=1

Bj − q

X

j=1

Aj ≥0.

The Wright generalized hypergeometric function [31] (seealso[12] ) qΨs[(α1, A1), ...,(αq, Aq) ; (β1, B1), ...,(βs, Bs) ;z] =q Ψs

h

(αi, Ai)_1,q; (βi, Bi)_1,s;z

i

is defined by

qΨsh(αi, Ai)_1,q; (βi, Bi)_1,s;z

i

=∞ n=0

q

i=1Γ (αi+nAi) s

Q

i=1

Γ (βi+nBi)

.z

n

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If Ai = 1(i= 1, ..., q) and Bi = 1 (i= 1, ..., s),we have the relationship: ΩqΨsh(αi,1)_1,q; (βi,1)_1,s;z

i

=qFs(α1, ..., αq;β1, ..., βs;z),

whereqFs(α1, ..., αq;β1, ..., βs;z) is the generalized hypergeometric function ( see for

details [9],[10],[11],[16] ) and

Ω = s

Q

i=1 Γ (βi) q

Q

i=1 Γ (αi)

.1.6 (6)

The Wright generalized hypergeometric functions were invoked in the geometric function theory ( see [8],[9],[10],[11],[25],[26] and [27] ).

By using the generalized hypergeometric function Dziok and Srivastava [10] in-troduced a linear operator. In [8] Dziok and Raina and in [2] Aouf and Dziok extended the linear operator by using Wright generalized hypergeometric function.

First we define a functionqΦps

h

(αi, Ai)1,q; (βi, Bi)1,s;z

i

by

qΦp_sh(αi, Ai)_1,q; (βi, Bi)_1,s;z

i

= Ωzp_qΨsh(αi, Ai)_1,q; (βi, Bi)_1,s;z

i

and consider the following linear operator

θp,q,s

h

(αi, Ai)_1,q; (βi, Bi)_1,s

i

:A(p)→A(p),1.7 (7) defined by the convolution

θp,q,s

h

(αi, Ai)_1,q; (βi, Bi)_1,s

i

f(z) =q Φp_sh(αi, Ai)_1,q; (βi, Bi)_1,s;z

i

∗f(z).

We observe that, for a function f(z) of the form (1.1),we have

θp,q,s

h

(αi, Ai)_1,q; (βi, Bi)_1,s

i

f(z) =zp+ ∞

X

n=p+1

Ωσn,p(α1)anzn,1.8 (8)

where Ω is given by (1.6) and σn,p(α1) is defined by

σn,p(α1) =

Γ (α1+A1(n−p))...Γ (αq+Aq(n−p)) Γ (β1+B1(n−p))...Γ (βs+Bs(n−p)) (n−p)!

.1.9 (9) If, for convenience, we write

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then one can easily verify from (1.8) that

zA1(θp,q,s[α1, A1, B1]f(z))′ =α1θp,q,s[α1+ 1, A1, B1]f(z)

−(α1−pA1)θp,q,s[α1, A1, B1]f(z) (A1>0).1.10 (10)

We note that for Ai = 1 (i= 1,2, ..., q) and Bi = 1 (i= 1,2, ..., s), we obtain

θp,q,s[α1,1,1]f(z) = Hp,q,s[α1]f(z), which was introduced and studied by Dziok and Srivastava [10].Also forf(z)∈A,the linear operatorθ1,q,s[α1, A1, B1] =θ[α1] was introduced by Dziok and Raina [8] and studied by Aouf and Dziok [2].

We note that, for f(z) ∈A(p), Ai = 1 (i= 1, ..., q), Bi = 1 (i= 1, ..., s), q = 2 and s= 1,we have:

(i) θp,2,1[a,1;c]f(z) =Lp(a, c)f(z) (a >0, c >0, p∈N) (seeSaitoh[28]); (ii) θp,2,1[µ+p, p;p]f(z) =Dµ+p−1f(z) (µ >−p, p∈N),whereDµ+p−1f(z) is the (µ+p−1)−the order Ruscheweyh derivative of a function f(z) ∈ A(p) (seeKumarandShukla[13]and[14]);

(iii) θp,2,1[1 +p,1; 1 +p−µ]f(z) = Ω(µ,p)z f(z), where the operator Ω(µ,p)z is defined by ( see Srivastava and Aouf [30] )

Ω(µ,p)_z f(z) = Γ (1 +p−µ) Γ (1 +p) z

µ_Dµ

zf(z) (0≤µ <1;p∈N), where Dzµ is the fractional derivative operator (see, f ordetails,[23]and[24] ) ;

(iv) θp,2,1[ν+p,1;ν+p+ 1]f(z) =Jν,p(f) (z), where Jν,p(f) (z) is the gener-alized Bernardi-Libera-Livingston-integral operator (see[3],[15]and[18] ), defined by

Jν,p(f) (z) =

ν+p zν

Z z

0

tν−₁

f(t)dt (ν >−p;p∈N) ;

(v)θp,2,1[p+ 1,1;n+p]f(z) =In,pf(z) (n∈Z, n >−p, p∈N),where the op-erator In,p was considered by Liu and Noor [17] ;

(vi) θp,2,1[λ+p, c;a]f(z) =Ipλ(a, c)f(z) a, c∈R\Z¯0, λ >−p, p∈N,where

Ipλ(a, c) is the Cho-Kwon-Srivastava operator [7].

The main object of the present paper is to find sufficient condition for certain normalized analytic functionsf(z) inU such thatθp,q,s[α1, A1, B1](f∗Ψ) (z)6= 0 and

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q1(z)≺

θp,q,s[α1, A1, B1](f ∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f∗Ψ) (z)

≺q2(z),

whereq1, q2 are given univalent functions in U and Φ(z) =zp+ ∞

P

n=p+1

λnzn, Ψ(z) =

zp₊ ∞

P

n=p+1

µnzn are analytic functions inU with λn ≥0, µn ≥ 0.Also, we obtain number of known results as special cases.

2. Definitions and Preliminaries

In order to prove our results, we shall make use of the following definition and lemmas.

Definition 1 [21]. Denote by Q, the set of all functions f that are analytic and injective on U \E(f),where

E(f) ={ξ∈∂U : lim

z→ξf(z) =∞}, and are such that f′

(ξ)6= 0 forξ∈∂U \E(f).

Lemma 1 [20].Let q be univalent in the unit disk U and θ and ϕ be analytic in a domain D containing q(U) with ϕ(w)6= 0 when w∈q(U).Set

ψ(z) =zq′

(z)ϕ(q(z))and h(z) =θ(q(z)) +ψ(z).it2.1 (11)

Suppose that

(i) ψ(z) is starlike univalent in U,

(ii) Renzh_ψ(z)′(z)o>0 for z∈U.

If p is analytic with p(0) =q(0), p(U)⊆D and θ(p(z)) +zp′

(z)ϕ(p(z))≺θ(q(z)) +zq′

(z)ϕ(q(z)), it2.2 (12)

then p(z)≺q(z) and q is the best dominant.

Lemma 2 [4].Let q be convex univalent in U and ϑand φbe analytic in a domain D containing q(U). Suppose that

(i) Re{ϑ′

(q(z))/φ(q(z))}>0 for z∈U and

(ii) ψ(z) =zq′

(z)φ(q(z)) is starlike univalent in U. If p(z) ∈H[q(0),1]∩Q, with p(U) ⊆D, and ϑ(p(z)) +zp′

(z)φ(p(z)) is univalent in U and

ϑ(q(z)) +zq′

(z)φ(q(z))≺ϑ(p(z)) +zp′

(z)φ(p(z)), it2.3 (13)

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3. Subordination results for analytic functions

Unless otherwise mentioned, we shall assume in the reminder of this paper that,

α1, A1, ..., αq, Aqandβ1, B1, ..., βs, Bs (q, s∈N) be positive real parameters, Φ, Ψ∈

A(p), p∈N, ξ1, ξ2, ξ3 ∈C and ξ4∈C∗=C\{0}.

Theorem 1. Let q be convex univalent in U with q(0) = 1. Further, assume that Re

ξ3

ξ4 +2ξ2

ξ4

q(z) +

1 +zq ′′

(z)

q′₍_z₎

>0 (z∈U).it3.1 (14)

If f ∈A(p) satisfies

̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)≺ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z),3.2 (15) where

̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

=

                

ξ1+ξ2

_θ

p,q,s[α1,A1,B1](f∗_Φ)(z) θp,q,s[α1+1,A1,B1](f∗_Ψ)(z)

2 +ξ3

_θ

p,q,s[α1,A1,B1](f∗_Φ)(z) θp,q,s[α1+1,A1,B1](f∗_Ψ)(z)

+ξ4

α1θp,q,s[α1+1,A1,B1](f ∗Φ)(z)

θp,q,s[α1,A1,B1](f∗_Φ)(z) −(α1+ 1)

θp,q,s[α1+2,A1,B1](f∗Ψ)(z) θp,q,s[α1+1,A1,B1](f∗_Ψ)(z) + 1

× θp,q,s[α1,A1,B1](f∗_Φ)(z) θp,q,s[α1+1,A1,B1](f∗Ψ)(z)

,

3.3

(16) then

θp,q,s[α1, A1, B1](f ∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f∗Ψ) (z)

≺q(z) and q is the best dominant.

Proof. Define the functionp(z) by

p(z) = θp,q,s[α1, A1, B1](f ∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f∗Ψ) (z)

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ξ1+ξ2

θp,q,s[α1, A1, B1](f∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f∗Ψ) (z)

2 +ξ3

θp,q,s[α1, A1, B1](f ∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f∗Ψ) (z)

+ξ4

α1

θp,q,s[α1+ 1, A1, B1](f∗Φ) (z)

θp,q,s[α1, A1, B1](f∗Φ) (z)

−(α1+ 1)

θp,q,s[α1+ 2, A1, B1](f∗Ψ) (z)

θp,q,s[α1+ 1, A1, B1](f∗Ψ) (z) + 1

×

θp,q,s[α1, A1, B1](f ∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f∗Ψ) (z)

=ξ1+ξ2p2(z) +ξ3p(z) +ξ4zp′(z).3.5 (18) By using (3.2) in (3.5), we have

ξ1+ξ2p2(z) +ξ3p(z) +ξ4zp′(z)≺ξ1+ξ2q2(z) +ξ3q(z) +ξ4zq′(z). By setting

θ(w) =ξ1+ξ2w2+ξ3w and ϕ(w) =ξ4,

it can be easily observed thatθ(w) is analytic in C,ϕ(w) is analytic in C∗

and that

ϕ(w)6= 0, w∈C∗

,then by using Lemma 1 we complete the proof of Theorem 1. Putting Φ(z) = Ψ(z) = ₁z−pz (or,µn=λn= 1, n≥p+ 1, p∈N) in Theorem 1, we obtain the following corollary:

Corollary 1. Let q be convex univalent in U with q(0) = 1and (3.1) holds true. If f ∈A(p) and satisfies

Z(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)≺ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z),3.6 (19) where

Z(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

=                 

ξ1+ξ2

_θ

p,q,s[α1,A1,B1]f(z)

θp,q,s[α1+1,A1,B1]f(z) 2

+ξ3

_θ

p,q,s[α1,A1,B1]f(z)

θp,q,s[α1+1,A1,B1]f(z)

+ξ4

α1θp,q,s_θ_p,q,s[α_[α1+1,A₁_,A₁_,B1,B₁_]f(z)1]f(z)−(α1+ 1)θ_θp,q,s_p,q,s[α_[α₁1+2,A_+1,A1₁_,B,B₁1]f(z)_]f(z)+ 1

× θp,q,s[α1,A1,B1]f(z)

θp,q,s[α1+1,A1,B1]f(z)

,

3.7

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then

θp,q,s[α1, A1, B1]f(z)

θp,q,s[α1+ 1, A1, B1]f(z)

≺q(z) and q is the best dominant.

Remark 1. Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s) and p = 1, in Corollary 1,we obtain the result obtained by Murugusundaramoorthy and Magesh [22, Corollary 2.6 ].

PuttingAi = 1 (i= 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2 ,s= 1 and α1 =

α2 =β1 = 1 in Theorem 1, hence we obtain the next result:

Corollary 2. Let q be convex univalent in U with q(0) = 1and (3.1) holds true. If f ∈A(p) and satisfies

B(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)≺ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z),3.8 (21) where

B(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

=

                

ξ1+ξ2

(f∗_Φ)(z)

z(f∗_Ψ)′_(z)+(1−_p)(f∗_Ψ)(z)

2 +ξ3

(f∗_Φ)(z)

z(f∗_Ψ)′_(z)+(1−_p)(f∗_Ψ)(z)

+ξ4

z(f∗_Φ)′_(z) (f∗_Φ)(z) −

z2_(f∗_Ψ)′′_(z)+z(3−_p)(f∗_Ψ)′_(z)+(1−_p)(3−_p)(f∗_Ψ)′_(z)

z(f∗_Ψ)′_(z)+(1−_p)(f∗_Ψ)(z) + (2−p)

× (f∗_Φ)(z)

z(f∗_Ψ)′_(z)+(1−_p)(f∗_Ψ)(z)

,

3.9

(22) then

(f∗Φ)(z)

z(f∗Ψ)′

(z) + (1−p)(f∗Ψ)(z) ≺q(z) and q is the best dominant.

Remark 2. (i) Putting p = 1 in Corollary 2, we obtain the result obtained by Murugusundaramoorthy and Magesh [22, Corollary 2.7 ];

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(iii) Putting p= 1and Φ(z) = Ψ(z) = ₁−zz (or, p= 1, µn=λn= 1and n≥2)in

Corollary 2, we obtain the result obtained by Murugusundaramoorthy and Magesh [22, Corollary 2.9 ].

Taking q(z) = 1 +Az1 +Bz (−1≤A < B≤1) in Theorem 1, we obtain the next result:

Corollary 3. Let q be convex univalent in U with q(0) = 1 and

Re

ξ3

ξ4 +2ξ2

ξ4

(1 +Az1 +Bz) +1−Bz 1 +Bz

>0 (z∈U). If f ∈A(p) ,−1≤A < B≤1 and

̥₍_f,_Φ_,_Ψ_{, ξ}₁_{, ξ}₂_{, ξ}₃_{, ξ}₄₎_≺_ξ₁₊_ξ₂_{(1 +}_Az_{1 +}_Bz₎2₍_z₎₊_ξ₃_{(1 +}_Az_{1 +}_Bz₎₊_ξ₄ z(A−B)

(1 +Bz)23.10 (23) where ̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is defined by (3.3), then

θp,q,s[α1, A1, B1](f∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f∗Ψ) (z)

≺1 +Az1 +Bz3.11 (24) and 1 +Az1 +Bz is the best dominant.

Remark 3. Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s) and p = 1,in

Corollary 3,we obtain the result obtained by Murugusundaramoorthy and Magesh [ 22, Corollary 2.11 ].

Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2, s = 1, α1 =

a, α2 = 1 and β1 = c (a, c > 0, p ∈ N) in Theorem 1, we obtain the following corollary:

Corollary 4. Let q be convex univalent in U with q(0) = 1 and (3.1) holds true.

K(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)≺ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z),3.12 (25) where

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=                 

ξ1+ξ2

_L

p(a,c)(f∗_Φ)(z) Lp(a+1,c)(f∗_Ψ)(z)

2 +ξ3

_L

p(a,c)(f∗_Φ)(z) Lp(a+1,c)(f∗_Ψ)(z)

+ξ4

aLp(a+1,c)(f∗_Φ)(z)

Lp(a,c)(f∗_Φ)(z) −(a+ 1)

Lp(a+2,c)(f∗_Ψ)(z) Lp(a+1,c)(f∗_Ψ)(z)+ 1

× Lp(a,c)(f∗Φ)(z) Lp(a+1,c)(f∗_Ψ)(z)

,

3.13

(26) then

Lp(a, c)(f ∗Φ) (z)

Lp(a+ 1, c)(f∗Ψ) (z)

≺q(z) and q is the best dominant of (3.12).

Remark 4. Putting p = 1 in Corollary 4, we obtain the result obtained by Muru-gusundaramoorthy and Magesh [22, Corollary 2.5 ].

µ+p (µ >−p, p∈N), α2 =β1=pin Theorem 1, we obtain the following corollary: Corollary 5. Let q be convex univalent in U with q(0) = 1 and (3.1) holds true.

M(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)≺ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z),3.14 (27)

where

M(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

=                 

ξ1+ξ2

Dµ+p−1_(f∗_Φ)(z) Dµ+p_(f∗_Ψ)(z)

2 +ξ3

Dµ+p−1_(f∗_Φ)(z) Dµ+p_(f∗_Ψ)(z)

+ξ4

(µ+p)_DDµ+pµ+p−(f1_(f∗Φ)(z)∗_Φ)(z)−(µ+p+ 1)

Dµ+p+1_(f∗_Ψ)(z) Dµ+p_(f∗_Ψ)(z) + 1

×Dµ+p−1(f∗_Φ)(z) Dµ+p_(f∗_Ψ)(z)

,

3.15

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Dµ+p−₁

(f∗Φ) (z)

Dµ+p₍_f _∗_{Ψ) (}_z₎ ≺q(z)

and q is the best dominant of (3.14).

Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2, s = 1, α1 = 1 +p, α2 = 1 andβ1= 1 +p−µ (0≤µ <1, p∈N),in Theorem 1, we obtain the following corollary:

Corollary 6. Let q be convex univalent in U with q(0) = 1and (3.1) holds true. If f ∈A(p) satisfies

N(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)≺ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z),3.16 (29)

where

N(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

=

                  

ξ1+ξ2

Ω(µ,p)z (f∗_Φ)(z) Ω(µ+1,p)z (f∗_Ψ)(z)

2 +ξ3

Ω(µ,p)z (f∗_Φ)(z) Ω(µ+1,p)z (f∗_Ψ)(z)

+ξ4

(1 +p)Ω(µ+1,p)z (f∗Φ)(z)

Ω(µ,p)z (f∗Φ)(z)

−(p+ 2)Ω(µ+2,p)z (f∗Ψ)(z)

Ω(µ+1,p)z (f∗Ψ)(z)

+ 1

×

Ω(µ,p)z (f∗Φ)(z) Ω(µ+1,p)z (f∗Ψ)(z)

,

3.17

(30)

then

Ω(µ,p)z (f ∗Φ) (z) Ω(µ+1,p)z (f∗Ψ) (z)

≺q(z)

ν +p (ν >−p, p ∈ N), α2 = 1 and β1 = ν+p+ 1 in Theorem 1, we obtain the following corollary:

Corollary 7. Let q be convex univalent in U with q(0) = 1 and (3.1) holds true.

(12)

where

L(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

=                 

ξ1+ξ2

_J

ν,p(f∗_Φ)(z) Jν+1,p(f∗_Ψ)(z)

2 +ξ3

_J

ν,p(f∗_Φ)(z) Jν+1,p(f∗_Ψ)(z)

+ξ4

(ν+p)Jν+1,p(f∗Φ)(z)

Jν,p(f∗_Φ)(z) −(ν+p+ 1)

Jν+2,p(f∗Ψ)(z) Jν+1,p(f∗_Ψ)(z) + 1

× Jν,p(f∗Φ)(z) Jν+1,p(f∗_Ψ)(z)

,

3.19

(32)

then

Jν,p(f ∗Φ) (z)

Jν+1,p(f ∗Ψ) (z) ≺q(z)

p+ 1, α2= 1 and β1=n+p (n∈Z, n >−p, p∈N) in Theorem 1, we obtain the following corollary:

Corollary 8. Let q be convex univalent in U with q(0) = 1 and (3.1) holds true. If f ∈A(p) satisfies

R(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)≺ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z),3.20 (33)

where

R(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

=                 

ξ1+ξ2

In,p(f∗_Φ)(z) In+1,p(f∗Ψ)(z)

2 +ξ3

In,p(f∗_Φ)(z) In+1,p(f∗Ψ)(z)

+ξ4

(p+ 1)In+1,p(f∗_Φ)(z)

In,p(f∗_Φ)(z) −(p+ 2)

In+2,p(f∗_Ψ)(z) In+1,p(f∗_Ψ)(z)+ 1

× In,p(f∗_Φ)(z) In+1,p(f∗_Ψ)(z)

,

3.21

(34)

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In,p(f∗Φ) (z)

In+1,p(f∗Ψ) (z) ≺q(z)

λ+p , α2=cand β1 =a a, c∈R\Z¯0, λ >−p, p∈N

,in Theorem 1, we obtain the following corollary:

Corollary 9. Let q be convex univalent in U with q(0) = 1 and (3.1) holds true. If f ∈A(p) satisfies

G(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)≺ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z),3.22 (35)

where

G(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

=

                  

ξ1+ξ2

_Iλ

p(a,c)(f∗Φ)(z)

Ipλ+1(a,c)(f∗_Ψ)(z)

2 +ξ3

_Iλ

p(a,c)(f∗Φ)(z)

Ipλ+1(a,c)(f∗_Ψ)(z)

+ξ4

(λ+p)Ipλ+1(a,c)(f∗Φ)(z) Iλ

p(a,c)(f∗Φ)(z) −(λ+p+ 1)

Ipλ+2(a,c)(f∗Ψ)(z) Ipλ+1(a,c)(f∗_Ψ)(z)+ 1

× Iλp(a,c)(f∗Φ)(z)

Ipλ+1(a,c)(f∗Ψ)(z)

,

3.23

(36)

then

Iλ

p (a, c) (f ∗Φ) (z)

Ipλ+1(a, c) (f∗Ψ) (z)

≺q(z)

4. Superordination results for analytic functions

Now, by applying Lemma 2, we prove the following theorem. Theorem 2. Let q be convex univalent in U with q(0) = 1and f ∈A(p), θp,q,s[α1,A1,B1](f∗_Φ)(z)

θp,q,s[α1+1,A1,B1](f∗_Ψ)(z) ∈H[q(0),1]∩Q.Further, assume that

Re

ξ3

ξ4 +2ξ2

ξ4

q(z)

(14)

and ̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) be univalent in U and satisfy

ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z)≺̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4),4.2 (38)

where ̥₍_f,_Φ_,_Ψ_{, ξ}₁_{, ξ}₂_{, ξ}₃_{, ξ}₄₎ _{is given by (3.3), then} q(z)≺ θp,q,s[α1, A1, B1](f∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f∗Ψ) (z)

and q is the best subordinant of (4.2).

Proof. Define the function p(z) by (3.4), with simple computation from (3.4), we get

̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) =ξ1+ξ2p2(z) +ξ3p(z) +ξ4 zp′(z), then

ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z)≺ξ1+ξ2p2(z) +ξ3p(z) +ξ4 zp′(z).4.3 (39)

By setting

υ(w) =ξ1+ξ2w2+ξ3w and ϕ(w) =ξ4, it is easy observed that υ(w) is analytic in C, ϕ(w) is analytic in C∗

and ϕ(w) 6= 0, w∈C∗

.Since q is convex and univalent function, it follows that

Reθ′

(q(z))ϕ(q(z)) = Re

ξ3

ξ4 +2ξ2

ξ4

q(z)

>0 (z∈U) and then, by using Lemma 2 we complete the proof of Theorem 2.

a, α2 = 1 and β1 =c(a, c >0) in Theorem 2, we obtain the following corollary: Corollary 10. Let q be convex univalent in U with q(0) = 1,(4.1) holds true and f ∈A(p), Lp(a,c)(f∗_Φ)(z)

Lp(a+1,c)(f∗_Ψ)(z) ∈H[q(0),1]∩Q.Further, assume that K(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)be

univalent in U which satisfying

ξ1+ξ2q2(z) +ξ3q(z) +ξ4 zq′(z)≺K(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4),4.4 (40)

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q(z)≺ Lp(a, c)(f∗Φ) (z)

Lp(a+ 1, c)(f∗Ψ) (z)

and q is the best subordinant of (4.4).

Remark 5. Putting p= 1in Corollary 10, we obtain the result obtained by Muru-gusundaramoorthy and Magesh [22, Corollary 2.13 ].

Remark 6. (i) Putting Ai = 1 (i= 1,2, ..., q), Bj = 1 (j= 1,2, ..., s), p= 1, q= 2

, s = 1, α1 = α2 = 1 and β1 = 1 in Theorem 2, we obtain the result obtained by

Murugusundaramoorthy and Magesh [22, Corollary 2.14 ];

(ii) Putting Ai = 1 (i= 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2 , s =p = α1 =

α2 = β1 = 1 and Φ(z) = Ψ(z) in Theorem 2, we obtain the result obtained by

Murugusundaramoorthy and Magesh [22, Corollary 2.15];

(iii) Putting Ai = 1 (i= 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2 , s=p=α1 =

α2 =β1 = 1and Φ(z) = Ψ(z) = ₁−zz in Theorem 2, we obtain the result obtained by

Murugusundaramoorthy and Magesh [22, Corollary 2.16].

Taking q(z) = 1 +Az1 +Bz (−1≤A < B≤1) in Theorem 2, we obtain the next result:

Corollary 11. Let q be convex univalent in U with q(0) = 1, f ∈A(p),

θp,q,s[α1,A1,B1](f∗_Φ)(z) θp,q,s[α1+1,A1,B1](f∗Ψ)(z) ∈H[q(0),1]∩Q and

Re

ξ3

ξ4 +2ξ2

ξ4

1 +Az1 +Bz

>0 (−1≤A < B≤1). Further, assume that ̥₍_f,_Φ_,_Ψ_{, ξ}₁_{, ξ}₂_{, ξ}₃_{, ξ}₄₎ _{be univalent in} _U _{and satisfy}

ξ1+ξ2(1 +Az1 +Bz)2(z)+ξ3(1 +Az1 +Bz)+ξ4

z(A−B)

(1 +Bz)2 ≺̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4),4.5 (41)

where ̥₍_f,_Φ_,_Ψ_{, ξ}₁_{, ξ}₂_{, ξ}₃_{, ξ}₄₎ _{is given by (3.3), then}

1 +Az1 +Bz≺ θp,q,s[α1, A1, B1](f∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f ∗Ψ) (z)

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Remark 7. Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s) and p = 1,in

Corollary 11,we obtain the result obtained by Murugusundaramoorthy and Magesh [22, Corollary 2.17 ].

Remark 8. (i) Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2, s= 1, α1 =µ+p (µ >−p, p∈N), α2 =β1 =p in Theorem 2, we obtain the

superordination result for the operator Dµ+p−1 ;

(ii) Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2, s = 1, α1 = 1 +p, α2 = 1and β1 = 1 +p−µ (0≤µ <1, p∈N),in Theorem 2, we obtain the

superordination result for the operator Ω(µ,p)z ;

(iii) Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2, s = 1, α1 =

ν +p (ν >−p, p ∈ N), α2 = 1 and β1 = ν+p+ 1 in Theorem 2, we obtain the

superordination result for the operator Jν,p;

(iv) Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2, s = 1, α1 =

p+ 1, α2= 1 and β1 =n+p (n∈Z, n >−p, p∈N)in Theorem 2, we obtain the

superordination result for the operator In,p;

(v) Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2, s = 1, α1 =

λ+p , α2 =cand β1 =a a, c∈R\Z¯0, λ >−p, p∈N

,in Theorem 2, we obtain the superordination result for the operator Ipλ(a, c).

4. Sandwich results

Combining Theorem 1 with Theorem 2, we get the following sandwich theorem. Theorem 3. Let q1, q2 be convex univalent in U.Suppose q1 and

q2 satisfies (4.1) and (3.1), respectively. If f ∈ A(p), θp,q,s[α1,A1,B1](f ∗_Φ)(z) θp,q,s[α1+1,A1,B1](f∗_Ψ)(z) ∈

H[q(0),1]∩Qand ̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)is univalent in U,which satisfying

ξ1+ξ2q21(z) +ξ3q1(z) +ξ4 zq1′(z)≺̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

≺ξ1+ξ2q22(z) +ξ3q2(z) +ξ4 zq2′(z),5.1 (42)

where ̥₍_f,_Φ_,_Ψ_{, ξ}₁_{, ξ}₂_{, ξ}₃_{, ξ}₄₎ _{is given by (3.3), then} q1(z)≺

θp,q,s[α1, A1, B1](f∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f ∗Ψ) (z)

≺q2(z)

and q1, q2 are, respectively, the best subordinant and dominant of (5.1).

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in Theorem 3, we obtain the following corollary:

Corollary 12. Let q1, q2 be convex univalent in U. Suppose q1 and q2 satisfies

(4.1) and (3.1), respectively. If f ∈ A(p), θp,q,s[α1,A1,B1](f∗Φ)(z)

θp,q,s[α1+1,A1,B1](f∗Ψ)(z) ∈ H[q(0),1]∩

Q and ̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is univalent in U,which satisfying

ξ1+ξ2(1 +A1z1 +B1z)2(z)+ξ3(1 +A1z1 +B1z)+ξ4

(A1−B1)z (1 +B1z)2

≺̥₍_f,_Φ_,_Ψ_{, ξ}₁_{, ξ}₂_{, ξ}₃_{, ξ}₄₎

≺ξ1+ξ2(1 +A2z1 +B2z)2+ξ3(1 +A2z1 +B2z) +ξ4 (A2−B2)z (1 +B2z)2

,5.2 (43)

where ̥(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is defined in (3.3),then 1 +A1z1 +B1z≺

θp,q,s[α1, A1, B1](f∗Φ) (z)

θp,q,s[α1+ 1, A1, B1](f ∗Ψ) (z)

≺1 +A2z1 +B2z

and 1 +A1z1 +B1z, 1 +A2z1 +B2z are, respectively, the best subordinant and

dominant of (5.2).

Remark 9. Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2 and

s = p = α1 = α2 = β1 = 1 in Corollary 12, we obtain the result obtained by

Murugusundaramoorthy and Magesh [22, Corollary 3.2 ].

a, α2 = 1 and β1 =c(a, c >0) in Theorem 3, we obtain the following corollary: Corollary 13. Let q1, q2 be convex univalent in U. Suppose q1 and q2 satisfies

(4.1) and (3.1), respectively. If f ∈A(p), Lp(a,c)(f∗_Φ)(z)

Lp(a+1,c)(f∗Ψ)(z) ∈H[q(0),1]∩Qand

K(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)is univalent in U ,where K(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)is defined in (3.13),then

ξ1+ξ2q12(z) +ξ3q1(z) +ξ4 zq1′(z)≺K(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

≺ξ1+ξ2q22(z) +ξ3q2(z) +ξ4 zq2′(z),5.3 (44)

implies

q1(z)≺

Lp(a, c)(f∗Φ) (z)

Lp(a+ 1, c)(f ∗Ψ) (z) ≺q2(z)

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µ+p (µ > −p, p ∈ N) and α2 = β1 = p in Theorem 3, we obtain the following corollary:

(4.1) and (3.1), respectively. If f ∈A(p), D_Dµ+pµ+p−_(f1(f∗_Ψ)(z)∗Φ)(z) ∈H[q(0),1]∩Q and

M(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is univalent in U, where M(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is defined

in (3.15),then

ξ1+ξ2q21(z) +ξ3q1(z) +ξ4 zq′1(z)≺M(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

≺ξ1+ξ2q22(z) +ξ3q2(z) +ξ4 zq2′(z),5.4 (45)

implies

q1(z)≺

Dµ+p−₁

(f ∗Φ) (z)

Dµ+p₍_f_∗_{Ψ) (}_z₎ ≺q2(z)

Putting Ai = 1 (i = 1,2, ..., q), Bj = 1 (j = 1,2, ..., s), q = 2, s = 1, α1 = 1 +p, α2 = 1 and β1 = 1 +p−µ (0≤µ <1, p∈N) in Theorem 3, we obtain the following corollary:

(4.1) and (3.1), respectively. If f ∈A(p), Ω(µ,p)z (f∗_Φ)(z)

Ω(µ+1,p)z (f∗_Ψ)(z) ∈H[q(0),1]∩Qand

N(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)is univalent in U,where N(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is defined in (3.17),then

ξ1+ξ2q12(z) +ξ3q1(z) +ξ4 zq1′(z)≺N(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

≺ξ1+ξ2q22(z) +ξ3q2(z) +ξ4 zq2′(z),5.5 (46)

implies

q1(z)≺

Ω(µ,p)z (f∗Φ) (z) Ω(µ+1,p)z (f ∗Ψ) (z)

≺q2(z)

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(4.1) and (3.1), respectively. If f ∈A(p), Jν,p(f∗_Φ)(z)

Jν+1,p(f∗_Ψ)(z) ∈H[q(0),1]∩Q and

L(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is univalent in U, where L(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is defined in (3.19),then

ξ1+ξ2q12(z) +ξ3q1(z) +ξ4 zq1′(z)≺L(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

≺ξ1+ξ2q22(z) +ξ3q2(z) +ξ4 zq2′(z),5.6 (47)

implies

q1(z)≺ Jν,p(f∗Φ) (z)

Jν+1,p(f∗Ψ) (z) ≺q2(z)

p+ 1, α2= 1 andβ1=n+p(n >−p, p∈N) in Theorem 3, we obtain the following corollary:

(4.1) and (3.1), respectively. If f ∈A(p), In,p(f∗Φ)(z)

In+1,p(f∗Ψ)(z) ∈H[q(0),1]∩Qand

R(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is univalent in U, where R(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is defined in (3.21),then

ξ1+ξ2q12(z) +ξ3q1(z) +ξ4 zq1′(z)≺R(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

≺ξ1+ξ2q22(z) +ξ3q2(z) +ξ4 zq2′(z),5.7 (48)

implies

q1(z)≺

In,p(f ∗Φ) (z)

In+1,p(f ∗Ψ) (z) ≺q2(z)

λ+p (λ > −p, p∈N), α2 =c and β1 =a a, c∈R\Z¯0, λ >−p

in Theorem 3, we obtain the following corollary:

(4.1) and (3.1), respectively. If f ∈A(p), Ipλ(a,c)(f∗Φ)(z)

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G(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is univalent in U, where G(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4) is defined in (3.23),then

ξ1+ξ2q12(z) +ξ3q1(z) +ξ4 zq1′(z)≺G(f,Φ,Ψ, ξ1, ξ2, ξ3, ξ4)

≺ξ1+ξ2q22(z) +ξ3q2(z) +ξ4 zq2′(z),5.8 (49)

implies

q1(z)≺

Iλ

p (a, c) (f∗Φ) (z)

Ipλ+1(a, c) (f ∗Ψ) (z)

≺q2(z)

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M. K. Aouf, A. Shamandy, A. O. Mostafa, S. M. Madian Department of Mathematics

Faculty of Science Mansoura University Mansoura 35516, Egypt.