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volume 7, issue 4, article 131, 2006.

Received 05 May, 2006;

accepted 04 August, 2006.

Communicated by:N.E. Cho

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Journal of Inequalities in Pure and Applied Mathematics

Aq-ANALOGUE OF AN INEQUALITY DUE TO KONRAD KNOPP

JIN-LIN LIU

Department of Mathematics Yangzhou University Yangzhou 225002, Jiangsu People’s Republic of China EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 132-06

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On Subordinations for Certain Multivalent Analytic Functions Associated with the Generalized

Hypergeometric Function Jin-Lin Liu

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J. Ineq. Pure and Appl. Math. 7(4) Art. 131, 2006

http://jipam.vu.edu.au

Abstract

The main object of the present paper is to investigate several interesting properties of a linear operatorH_p,q,s(α_i)associated with the generalized hypergeometric function.

2000 Mathematics Subject Classification:30C45, 26A33.

Key words: Analytic functions; The generalized hypergeometric function; Differential subordination; Univalent functions; Hadamard product (or convolution).

The present research is partly supported by Jiangsu Gaoxiao Natural Science Foun- dation (04KJB110154).

1. Introduction

LetA(p)denote the class of functions of the form (1.1) f(z) = z^p+

∞

X

n=p+1

a_nzⁿ, (p∈N ={1,2,3, . . .})

which are analytic in the open unit diskU ={z:z ∈C and|z|<1}.

Let f(z) and g(z) be analytic in U. Then we say that the function g(z) is subordinate to f(z)if there exists an analytic function w(z) inU such that

|w(z)| < 1 (forz ∈ U) and g(z) = f(w(z)). This relation is denotedg(z) ≺ f(z). In casef(z)is univalent inU we have that the subordinationg(z)≺f(z) is equivalent tog(0) =f(0)andg(U)⊂f(U).

For analytic functions f(z) =

∞

X

n=0

a_nzⁿ and g(z) =

∞

X

n=0

b_nzⁿ,

byf∗gwe denote the Hadamard product or convolution off andg, defined by

(1.2) (f∗g)(z) =

∞

X

n=0

a_nb_nzⁿ= (g∗f)(z).

Next, for real parametersA andB such that−1 ≤ B < A ≤ 1, we define the function

(1.3) h(A, B;z) = 1 +Az

1 +Bz (z ∈U).

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It is well known thath(A, B;z)for−1 ≤ B ≤ 1is the conformal map of the unit disk onto the disk symmetrical with respect to the real axis having the center (1−AB)/(1−B²)and the radius(A−B)/(1−B²)forB 6=∓1. The boundary circle cuts the real axis at the points(1−A)/(1−B)and(1 +A)/(1 +B).

For complex parametersα₁, . . . , α_qandβ₁, . . . , β_s(β_j 6= 0,−1,−2, . . .;j = 1, . . . , s), we define the generalized hypergeometric function _qF_s(α₁, . . . , α_q; β₁, . . . , β_s;z)by

qFs(α1, . . . , αq;β1, . . . , βs;z) =

∞

X

n=0

(α₁)_n· · ·(α_q)_n (β₁)_n· · ·(β_s)_n · zⁿ

n!

(q≤s+ 1;q, s∈N₀ =N ∪ {0};z ∈U), (1.4)

where(x)_n is the Pochhammer symbol, defined, in terms of the Gamma func- tionΓ, by

(x)_n= Γ(x+n) Γ(x) =

1 (n = 0),

x(x+ 1)· · ·(x+n−1) (n ∈N).

Corresponding to a functionFp(α₁, . . . , α_q;β₁, . . . , β_s;z)defined by Fp(α₁, . . . , α_q;β₁, . . . , β_s;z) =z^p_qF_s(α₁, . . . , α_q;β₁, . . . , β_s;z), we consider a linear operator

H_p(α₁, . . . , α_q;β₁, . . . , β_s) :A(p)→A(p), defined by the convolution

(1.5) Hp(α1, . . . , αq;β1, . . . , βs)f(z) = Fp(α1, . . . , αq;β1, . . . , βs;z)∗f(z).

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For convenience, we write

(1.6) H_p,q,s(α_i) = H_p(α₁, . . . , α_i, . . . , α_q;β₁, . . . , β_s) (i= 1,2, . . . , q).

Thus, after some calculations, we have

z(H_p,q,s(α_i)f(z))⁰ =α_iH_p,q,s(α_i+ 1)f(z)−(α_i−p)H_p,q,s(α_i)f(z) (i= 1,2, . . . , q).

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It should be remarked that the linear operator H_p,q,s(α_i) (i = 1,2, . . . , q) is a generalization of many operators considered earlier. For q = 2 and s = 1 Carlson and Shaffer studied this operator under certain restrictions on the parameters α₁, α₂ and β₁ in [1]. A more general operator was studied by Pon- nusamy and Rønning [13]. Also, many interesting subclasses of analytic functions, associated with the operator H_p,q,s(α_i) (i = 1,2, . . . , q) and its many special cases, were investigated recently by (for example) Dziok and Srivastava [2, 3, 4], Gangadharan et al. [5], Liu [7], Liu and Srivastava [8, 9] and others (see also [6,12,15,16,17]).

In the present sequel to these earlier works, we shall use the method of differential subordination to derive several interesting properties and characteristics of the operatorH_p,q,s(α_i) (i= 1,2, . . . , q).

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2. Main Results

We begin by recalling each of the following lemmas which will be required in our present investigation.

Lemma 2.1 (see [10]). Leth(z)be analytic and convex univalent inU,h(0) = 1and letg(z) = 1 +b₁z+b₂z²+· · · be analytic inU. If

(2.1) g(z) +zg⁰(z)/c≺h(z) (z ∈U;c6= 0), then forRec≥0,

(2.2) g(z)≺ c

z^c Z z

0

t^c−1h(t)dt.

Lemma 2.2 (see [14]). The function(1−z)^γ ≡ e^γ^log(1−z),γ 6= 0, is univalent inU if and only ifγis either in the closed disk|γ−1| ≤1or in the closed disk

|γ+ 1| ≤1.

Lemma 2.3 (see [11]). Letq(z)be univalent in U and let θ(w) and φ(w)be analytic in a domain Dcontainingq(U)with φ(w) 6= 0whenw ∈ q(U). Set Q(z) = zq⁰(z)φ(q(z)), h(z) = θ(q(z)) +Q(z)and suppose that

1. Q(z)is starlike (univalent) inU; 2. Re_zh0(z)

Q(z)

= Re_θ0(q(z))

φ(q(z)) +^zQ_Q(z)⁰^(z)

>0 (z∈U).

Ifp(z)is analytic inU, withp(0) =q(0), p(U)⊂D, and

θ(p(z)) +zp⁰(z)φ(p(z))≺θ(q(z)) +zq⁰(z)φ(q(z)) =h(z), thenp(z)≺q(z)andq(z)is the best dominant.

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We now prove our first result given by Theorem2.4below.

Theorem 2.4. Let αi >0 (i = 1,2, . . . , q), λ > 0, and−1 ≤ B < A ≤ 1. If f(z)∈A(p)satisfies

(2.3) (1−λ)Hp,q,s(αi)f(z)

z^p +λHp,q,s(αi+ 1)f(z)

z^p ≺h(A, B;z), then

(2.4) Re

H_p,q,s(α_i)f(z) z^p

_m¹!

>

α_i λ

Z 1

0

u^αi^λ⁻¹

1−Au 1−Bu

du

_m¹

(m≥1).

The result is sharp.

Proof. Let

(2.5) g(z) = Hp,q,s(αi)f(z) z^p

forf(z)∈ A(p). Then the functiong(z) = 1 +b₁z +· · · is analytic in U. By making use of (1.7) and (2.5), we obtain

(2.6) H_p,q,s(α_i+ 1)f(z)

z^p =g(z) + zg⁰(z) α_i .

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From (2.3), (2.5) and (2.6) we get

(2.7) g(z) + λ

α_izg⁰(z)≺h(A, B;z).

Now an application of Lemma2.1leads to

(2.8) g(z)≺ α_i

λz⁻^αi^λ Z 1

0

t^αi^λ⁻¹

1 +At 1 +Bt

dt

or

(2.9) H_p,q,s(α_i)f(z)

z^p = α_i

λ Z 1

0

u^αi^λ⁻¹

1 +Auw(z) 1 +Buw(z)

du, wherew(z)is analytic inU withw(0) = 0and|w(z)|<1(z ∈U).

In view of−1≤B < A≤1andα_i >0, it follows from (2.9) that (2.10) Re

> α_i λ

Z 1

0

u^αi^λ⁻¹

1−Au 1−Bu

du (z ∈U).

Therefore, with the aid of the elementary inequalityRe(w^1/m)≥(Rew)^1/mfor Rew >0andm ≥1, the inequality (2.4) follows directly from (2.10).

To show the sharpness of (2.4), we takef(z)∈A(p)defined by H_p,q,s(α_i)f(z)

z^p = α_i

λ Z 1

0

u^αi^λ⁻¹

1 +Auz 1 +Buz

du.

For this function, we find that (1−λ)H_p,q,s(α_i)f(z)

z^p +λH_p,q,s(α_i+ 1)f(z)

z^p = 1 +Az

1 +Bz

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and

Hp,q,s(αi)f(z)

z^p → αi

λ Z 1

0

u^αi^λ⁻¹

1−Au 1−Bu

du asz → −1.

Hence the proof of the theorem is complete.

Next we prove our second theorem.

Theorem 2.5. Letα_i >0 (i= 1,2, . . . , q), and0≤ρ <1. Letγ be a complex number withγ 6= 0and satisfy either|2γ(1−ρ)αi−1| ≤1or|2γ(1−ρ)αi+1| ≤ 1 (i= 1,2, . . . , q). Iff(z)∈A(p)satisfies the condition

(2.11) Re

H_p,q,s(α_i+ 1)f(z) H_p,q,s(α_i)f(z)

> ρ (z ∈U;i= 1,2, . . . , q),

then

(2.12)

γ

≺ 1

(1−z)^{2γ(1−ρ)α}ⁱ =q(z) (z ∈U;i= 1,2, . . . , q),

whereq(z)is the best dominant.

Proof. Let

(2.13) p(z) =

γ

(z ∈U;i= 1,2, . . . , q).

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Then, by making use of (1.7), (2.11) and (2.13), we have (2.14) 1 + zp⁰(z)

γα_ip(z) ≺ 1 + (1−2ρ)z

1−z (z ∈U).

If we take

q(z) = 1

(1−z)^{2γ(1−ρ)α}ⁱ, θ(w) = 1 and φ(w) = 1 γα_iw,

thenq(z)is univalent by the condition of the theorem and Lemma2.2. Further, it is easy to show thatq(z), θ(w)andφ(w)satisfy the conditions of Lemma2.3.

Since

Q(z) =zq⁰(z)φ(q(z)) = 2(1−ρ)z 1−z is univalent starlike inU and

h(z) =θ(q(z)) +Q(z) = 1 + (1−2ρ)z 1−z .

It may be readily checked that the conditions (1) and (2) of Lemma 2.3 are satisfied. Thus the result follows from (2.14) immediately. The proof is complete.

Corollary 2.6. Let α_i > 0 (i = 1,2, . . . , q) and0 ≤ ρ < 1. Let γ be a real number andγ ≥1. Iff(z)∈A(p)satisfies the condition (2.11), then

Re

_2γ(1−ρ)¹

αi >2^−1/γ (z ∈U;i= 1,2, . . . , q).

The bound2^−1/γ is the best possible.

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References

[1] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hyperge- ometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.

[2] J. DZIOKANDH.M. SRIVASTAVA, Classes of analytic functions associ- ated with the generalized hypergeometric function, Appl. Math. Comput., 103 (1999), 1–13.

[3] J. DZIOK AND H.M. SRIVASTAVA, Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. Stud. Contemp. Math., 5 (2002), 115–125.

[4] J. DZIOKAND H.M. SRIVASTAVA, Certain subclasses of analytic func- tions associated with the generalized hypergeometric function, Integral Transform. Spec. Funct., 14 (2003), 7–18.

[5] A. GANGADHARAN, T.N. SHANMUGAM AND H.M. SRIVASTAVA, Generalized hypergeometric functions associated withk-uniformly convex functions, Comput. Math. Appl., 44 (2002), 1515–1526.

[6] Y.C. KIM AND H.M. SRIVASTAVA, Fractional integral and other linear operators associated with the Gaussian hypergeometric function, Complex Variables Theory Appl., 34 (1997), 293–312.

[7] J.-L. LIU, Strongly starlike functions associated with the Dziok-Srivastava operator, Tamkang J. Math., 35 (2004), 37–42.

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[8] J.-L. LIU ANDH.M.SRIVASTAVA, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Math. Comput. Modelling, 38 (2004), 21–34.

[9] J.-L. LIU AND H.M. SRIVASTAVA, Certain properties of the Dziok- Srivastava operator, Appl. Math. Comput., 159 (2004), 485–493.

[10] S.S. MILLER AND P.T. MOCANU, Differential subordinations and uni- valent functions, Michigan Math. J., 28 (1981), 157–171.

[11] S.S. MILLER ANDP.T. MOCANU, On some classes of first order differ- ential subordination, Michigan Math. J., 32 (1985), 185–195.

[12] S. OWA ANDH.M. SRIVASTAVA, Univalent and starlike generalized hy- pergeometric functions, Canad. J. Math., 39 (1987), 1057–1077.

[13] S. PONNUSAMY AND F. RØNNING, Duality for Hadamard products applied to certain integral transforms, Complex Variables: Theory Appl., 32 (1997), 263–287.

[14] M.S. ROBERTSON, Certain classes of starlike functions, Michigan Math.

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[15] H.M. SRIVASTAVA AND S. OWA, Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of ana- lytic functions, Nagoya Math. J., 106 (1987), 1–28.

[16] H.M. SRIVASTAVAANDS. OWA (Eds.), Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press (Eills Horwood Limited,

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Chichester), John Wiley and Sons, New York,Chichester, Brisbane, and Toronto, (1989).

[17] H.M. SRIVASTAVA AND S. OWA (Eds.), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, (1992).