A STUDY ON STARLIKE AND CONVEX PROPERTIES FOR HYPERGEOMETRIC FUNCTIONS

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Study on Starlike and Convex Properties

A. O. Mostafa vol. 10, iss. 3, art. 87, 2009

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A STUDY ON STARLIKE AND CONVEX

PROPERTIES FOR HYPERGEOMETRIC FUNCTIONS

A. O. MOSTAFA

Department of Mathematics Faculty of Science

Mansoura University Mansoura 35516, Egypt EMail:adelaeg254@yahoo.com

Received: 07 May, 2008

Accepted: 21 August, 2009

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C45.

Key words: Starlike, Convex, Hypergeometric function, Integral operator.

Abstract: The objective of the present paper is to give some characterizations for a (Gaus- sian) hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function.

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1. Introduction

LetT be the class consisting of functions of the form:

(1.1) f(z) = z−

∞

X

n=2

a_nzⁿ (a_n>0),

which are analytic and univalent in the open unit discU ={z :|z|<1}.LetT(λ, α) be the subclass ofT consisting of functions which satisfy the condition:

(1.2) Re

zf⁰(z)

λzf⁰(z) + (1−λ)f(z)

> α, for someα(0≤α <1), λ(0≤λ <1)and for allz∈U.

Also, letC(λ, α)denote the subclass of T consisting of functions which satisfy the condition:

(1.3) Re

f⁰(z) +zf⁰⁰(z) f⁰(z) +λzf⁰⁰(z)

> α, for someα(0≤α <1), λ(0≤λ <1)and for allz∈U.

From(1.2)and(1.3), we have

(1.4) f(z)∈C(λ, α)⇔zf⁰(z)∈T(λ, α).

We note thatT(0, α) =T^∗(α),the class of starlike functions of orderα (0≤α <1) and C(0, α) = C(α), the class of convex functions of orderα (0 ≤ α < 1) (see Silverman [6]).

LetF(a, b;c;z)be the (Gaussian ) hypergeometric function defined by

(1.5) F(a, b;c;z) =

∞

X

n=0

(a)n(b)n

(c)_n(1)_nzⁿ,

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wherec6= 0,−1,−2, ...,and(θ)_nis the Pochhammer symbol defined by (θ)_n =

( 1, n= 0

θ(θ+ 1)· · ·(θ+n−1) n∈N ={1,2, ...}.

We note that F(a, b;c; 1) converges for Re(c− a −b) > 0 and is related to the Gamma function by

(1.6) F(a, b;c; 1) = Γ(c)Γ(c−a−b) Γ(c−a)Γ(c−b).

Silverman [7] gave necessary and sufficient conditions forzF(a, b;c;z)to be in T^∗(α)and C(α), also examining a linear operator acting on hypergeometric functions. For other interesting developments onzF(a, b;c;z)in connection with various subclasses of univalent functions, the reader can refer the to works of Carlson and Shaffer [2], Merkes and Scott [4], Ruscheweyh and Singh [5] and Cho et al. [3].

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2. Main Results

To establish our main results, we need the following lemma due to Altintas and Owa [1].

Lemma 2.1.

(i) A functionf(z)defined by(1.1)is in the classT(λ, α)if and only if (2.1)

∞

X

n=2

(n−λαn−α+λα)a_n ≤1−α.

(ii) A functionf(z)defined by(1.1)is in the classC(λ, α)if and only if (2.2)

∞

X

n=2

n(n−λαn−α+λα)a_n≤1−α.

Theorem A.

(i) Ifa, b >−1, c > 0andab <0,thenzF(a, b;c;z)is inT(λ, α)if and only if (2.3) c > a+b+ 1− (1−λα)ab

1−α .

(ii) If a, b > 0 andc > a+b+ 1, thenF1(a, b;c;z) = z[2−F(a, b;c;z)] is in T(λ, α)if and only if

(2.4) Γ(c)Γ(c−a−b) Γ(c−a)Γ(c−b)

1 + (1−λα)ab (1−α)(c−a−b−1)

≤2.

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Proof. (i)Since

zF(a, b;c;z) =z+ ab c

∞

X

n=2

(a+ 1)n−2(b+ 1)n−2

(c+ 1)n−2(1)n−1

zⁿ (2.5)

=z−

ab c

∞

X

n=2

(a+ 1)n−2(b+ 1)n−2

(c+ 1)n−2(1)n−1

zⁿ,

according to(i)of Lemma2.1, we must show that (2.6)

∞

X

n=2

(n−λαn−α+λα)(a+ 1)n−2(b+ 1)n−2

(c+ 1)n−2(1)n−1

≤

c ab

(1−α). Note that the left side of(2.6)diverges ifc < a+b+ 1. Now

∞

X

n=2

(n−λαn−α+λα)(a+ 1)_n−2(b+ 1)_n−2 (c+ 1)n−2(1)n−1

= (1−λα)

∞

X

n=0

(n+ 1)(a+ 1)_n(b+ 1)_n (c+ 1)n(1)n+1

+ (1−α)

∞

X

n=0

(a+ 1)_n(b+ 1)_n (c+ 1)n(1)n+1

= (1−λα)

∞

X

n=0

(a+ 1)_n(b+ 1)_n

(c+ 1)_n(1)_n +(1−α)c ab

∞

X

n=1

(a)_n(b)_n (c)_n(1)_n

= (1−λα)Γ(c+ 1)Γ(c−a−b−1)

Γ(c−a)Γ(c−b) +(1−α)c ab

Γ(c)Γ(c−a−b) Γ(c−a)Γ(c−b)−1

.

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Hence,(2.6)is equivalent to (2.7) Γ(c+ 1)Γ(c−a−b−1)

Γ(c−a)Γ(c−b)

(1−λα) + (1−α)(c−a−b−1) ab

≤(1−α)h

c ab

+ c

ab i

= 0.

Thus,(2.7)is valid if and only if

(1−λα) + (1−α)(c−a−b−1)

ab ≤0,

or equivalently,

c>a+b+ 1− (1−λα)ab 1−α . (ii)Since

F₁(a, b;c;z) =z−

∞

X

n=2

(a)n−1(b)n−1

(c)n−1(1)n−1

zⁿ, by(i)of Lemma2.1, we need only to show that

∞

X

n=2

(n−λαn−α+λα)(a)n−1(b)n−1

(c)n−1(1)n−1

≤1−α.

Now,

∞

X

n=2

(n−λαn−α+λα)(a)n−1(b)n−1

(c)n−1(1)n−1

(2.8)

=

∞

X

n=2

[(n−1)(1−λα) + (1−α)](a)n−1(b)n−1

(c)n−1(1)n−1

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= (1−λα)

∞

X

n=1

n(a)_n(b)_n

(c)_n(1)_n + (1−α)

∞

X

n=1

(a)_n(b)_n (c)_n(1)_n

= (1−λα)

∞

X

n=1

(a)_n(b)_n (c)_n(1)n−1

+ (1−α)

∞

X

n=1

(a)_n(b)_n (c)_n(1)_n. Noting that(θ)_n=θ((θ+ 1)n−1then,(2.8)may be expressed as

(1−λα)ab c

∞

X

n=1

(a+ 1)n−1(b+ 1)n−1

(c+ 1)_n−1(1)_n−1 + (1−α)

∞

X

n=1

(a)_n(b)_n (c)_n(1)_n

= (1−λα)ab c

∞

X

n=0

(a+ 1)_n(b+ 1)_n

(c+ 1)_n(1)_n + (1−α)

" _∞ X

n=0

(a)_n(b)_n (c)_n(1)_n −1

#

= (1−λα)ab c

Γ(c+ 1)Γ(c−a−b−1)

Γ(c−a)Γ(c−b) + (1−α)

Γ(c)Γ(c−a−b) Γ(c−a)Γ(c−b)−1

= Γ(c)Γ(c−a−b) Γ(c−a)Γ(c−b)

(1−α) + ab(1−λα) (c−a−b−1)

−(1−α).

But this last expression is bounded above by1−αif and only if(2.4)holds.

Theorem B.

(i) Ifa, b >−1,ab < 0,andc > a+b+ 2,thenzF(a, b;c;z)is inC(λ, α)if and only if

(2.9) (1−λα)(a)₂(b)₂ + (3−2λα−α)ab(c−a−b−2)

+ (1−α)(c−a−b−2)₂ >0.

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(ii) If a, b > 0 and c > a+b+ 2, thenF₁(a, b;c;z) = z[2−F(a, b;c;z)] is in C(λ, α)if and only if

(2.10) Γ(c)Γ(c−a−b) Γ(c−a)Γ(c−b)

1 + (1−λα)(a)₂(b)₂ (1−α)(c−a−b−2)₂ +

3−2λα−α 1−α

ab c−a−b−1

≤2.

Proof. (i) SincezF(a, b;c;z)has the form (2.5), we see from(ii) of Lemma 2.1, that our conclusion is equivalent to

(2.11)

∞

X

n=2

n(n−λαn−α+λα)(a+ 1)n−2(b+ 1)n−2

(c+ 1)n−2(1)n−1

≤

c ab

(1−α). Note that forc > a+b+ 2, the left side of(2.11)converges. Writing

(n+ 2)[(n+ 2)(1−λα)−α(1−λ)]

= (n+ 1)²(1−λα) + (n+ 1)(2−α−λα) + (1−α), we see that

∞

X

n=0

(n+ 2)[(n+ 2)(1−λα)−α(1−λ)](a+ 1)_n(b+ 1)_n (c+ 1)n(1)n+1

= (1−λα)

∞

X

n=0

(n+ 1)²(a+ 1)_n(b+ 1)_n (c+ 1)_n(1)_n+1 + (2−α−λα)

∞

X

n=0

(n+ 1)(a+ 1)n(b+ 1)n

(c+ 1)_n(1)_n+1 + (1−α)

∞

X

n=0

(a+ 1)n(b+ 1)n

(c+ 1)_n(1)_n+1

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= (1−λα)

∞

X

n=0

(n+ 1)(a+ 1)_n(b+ 1)_n (c+ 1)_n(1)_n + (2−α−λα)

∞

X

n=0

(a+ 1)_n(b+ 1)_n

(c+ 1)_n(1)_n + (1−α)

∞

X

n=0

(a+ 1)_n(b+ 1)_n (c+ 1)_n(1)_n+1

= (1−λα)

∞

X

n=0

n(a+ 1)_n(b+ 1)_n (c+ 1)n(1)n

+ (3−α−2λα)

∞

X

n=0

(a+ 1)_n(b+ 1)_n (c+ 1)n(1)n

+ (1−α)

∞

X

n=1

(a+ 1)n−1(b+ 1)n−1

(c+ 1)_n−1(1)_n

= (1−λα)(a+ 1)(b+ 1) (c+ 1)

∞

X

n=0

(a+ 2)_n(b+ 2)_n (c+ 2)_n(1)_n + (3−α−2λα)

∞

X

n=0

(a+ 1)_n(b+ 1)_n

(c+ 1)_n(1)_n + (1−α) c ab

∞

X

n=1

(a)_n(b)_n (c)_n(1)_n

= Γ(c+ 1)Γ(c−a−b−2) Γ(c−a)Γ(c−b)

−(1−λα)(a+ 1)(b+ 1) +(3−α−2λα)(c−a−b−2) + (1−α)

ab (c−a−b−2)₂

− (1−α)c ab . This last expression is bounded above by

_ab^c

(1−α)if and only if (1−λα)(a+1)(b+1)+(3−α−2λα)(c−a−b−2)+(1−α)

ab (c−a−b−2)₂ ≤0, which is equivalent to(2.9).

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(ii)In view of(ii)of Lemma2.1, we need to show that

∞

X

n=2

n(n−λαn−α+λα)(a)n−1(b)n−1

(c)n−1(1)n−1

≤(1−α). Now

∞

X

n=2

n(n−λαn−α+λα)(a)n−1(b)n−1

(c)n−1(1)n−1

(2.12)

=

∞

X

n=0

(n+ 2)[(n+ 2)(1−λα)−α(1−λ)](a)_n+1(b)_n+1 (c)_n+1(1)_n+1

= (1−λα)

∞

X

n=0

(n+2)²(a)_n+1(b)_n+1

(c)_n+1(1)_n+1−α(1−λ)

∞

X

n=0

(n+2)(a)_n+1(b)_n+1 (c)_n+1(1)_n+1. Writing(n+ 2) = (n+ 1) + 1,we have

∞

X

n=0

(n+ 2)(a)_n+1(b)_n+1 (c)_n+1(1)_n+1 =

∞

X

n=0

(n+ 1)(a)_n+1(b)_n+1 (c)_n+1(1)_n+1 +

∞

X

n=0

(a)_n+1(b)_n+1 (c)_n+1(1)_n+1 (2.13)

=

∞

X

n=0

(a)_n+1(b)_n+1 (c)_n+1(1)_n +

∞

X

n=0

(a)_n+1(b)_n+1 (c)_n+1(1)_n+1 and

∞

X

n=0

(n+ 2)²(a)_n+1(b)_n+1 (c)_n+1(1)_n+1 (2.14)

=

∞

X

n=0

(n+ 1)(a)n+1(b)n+1

(c)_n+1(1)_n + 2

∞

X

n=0

(a)n+1(b)n+1

(c)_n+1(1)_n +

∞

X

n=0

(a)n+1(b)n+1

(c)_n+1(1)_n+1

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=

∞

X

n=1

(a)_n+1(b)_n+1 (c)_n+1(1)n−1

+ 3

∞

X

n=0

(a)_n+1(b)_n+1 (c)_n+1(1)_n +

∞

X

n=1

(a)_n(b)_n (c)_n(1)_n. Substituting(2.13)and(2.14)into the right side of(2.12), yields (2.15) (1−λα)

∞

X

n=0

(a)_n+2(b)_n+2

(c)_n+2(1)_n + (3−2λα−α)

∞

X

n=0

(a)_n+1(b)_n+1 (c)_n+1(1)_n + (1−α)

∞

X

n=1

(a)_n(b)_n (c)_n(1)_n. Since(a)_n+k= (a)_k(a+k)_n,we may write(2.15)as

Γ(c)Γ(c−a−b) Γ(c−a)Γ(c−b)

(1−λα)(a)₂(b)₂

(c−a−b−2)₂ +(3−2λα−α)ab

(c−a−b−1) + (1−α)

−(1−α).

By a simplification, we see that the last expression is bounded above by(1−α)if and only if(2.10)holds.

Puttingλ= 0in(i)of TheoremB, we have:

Corollary 2.2. If a, b > −1, ab < 0, and c > a+b + 2,then zF(a, b;c;z) is in C(α)if and only if

(a)₂(b)₂+ (3−α)ab(c−a−b−2) + (1−α)(c−a−b−2)₂ >0.

Remark 1. Corollary2.2, corrects the result obtained by Silverman [7, Theorem 4].

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3. An Integral Operator

In the theorems below, we obtain similar results in connection with a particular integral operatorG(a, b;c;z)acting onF(a, b;c;z)as follows:

(3.1) G(a, b;c;z) =

Z z

0

F(a, b;c;t)dt.

Theorem C. Let a, b > −1, ab < 0 and c > max{0, a +b}. Then G(a, b;c;z) defined by(3.1)is inT(λ, α)if and only if

(3.2) Γ(c+ 1)Γ(c−a−b) Γ(c−a)Γ(c−b)

(1−λα)

ab − α(1−λ)(c−a−b) (a−1)₂(b−1)₂

+α(1−λ)(c−1)₂ (a−1)₂(b−1)₂ ≤0.

Proof. Since

G(a, b;c;z) = z−

ab c

∞

X

n=2

(a+ 1)n−2(b+ 1)n−2

(c+ 1)n−2(1)n

zⁿ,

by(i)of Lemma2.1, we need only to show that

∞

X

n=2

(n−λαn−α+λα)(a+ 1)n−2(b+ 1)n−2

(c+ 1)n−2(1)_n ≤

c ab

(1−α).

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Now

∞

X

n=2

[n(1−λα)−α(1−λ)](a+ 1)_n−2(b+ 1)_n−2 (c+ 1)n−2(1)_n

= (1−λα)

∞

X

n=0

(n+ 2)(a+ 1)_n(b+ 1)_n (c+ 1)n(1)n+2

−α(1−λ)

∞

X

n=0

(a+ 1)_n(b+ 1)_n (c+ 1)n(1)n+2

= (1−λα)

∞

X

n=0

(a+ 1)_n(b+ 1)_n

(c+ 1)_n(1)_n+1 −α(1−λ)

∞

X

n=1

(a+ 1)n−1(b+ 1)n−1

(c+ 1)_n−1(1)_n+1

= (1−λα)

∞

X

n=1

(a+ 1)n−1(b+ 1)n−1

(c+ 1)n−1(1)_n −α(1−λ) c ab

∞

X

n=1

(a)_n(b)_n (c)_n(1)_n+1

= (1−λα) c ab

∞

X

n=1

(a)_n(b)_n

(c)_n(1)_n −α(1−λ) c ab

∞

X

n=1

(a)_n(b)_n (c)_n(1)_n+1

= (1−λα) c ab

" _∞ X

n=0

(a)_n(b)_n (c)n(1)n

−1

#

−α(1−λ) (c−1)₂ (a−1)₂(b−1)₂

∞

X

n=2

(a−1)_n(b−1)_n (c−1)_n(1)_n

= Γ(c+ 1)Γ(c−a−b) Γ(c−a)Γ(c−b)

(1−λα)

ab − α(1−λ)(c−a−b) (a−1)₂(b−1)₂

+ α(1−λ)(c−1)₂

(a−1)₂(b−1)₂ − (1−α)c ab , which is bounded above by(1−α)

_ab^c

if and only if(3.2)holds.

Now, we observe that G(a, b;c;z) ∈ C(λ, α) if and only if zF(a, b;c;z) ∈

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T(λ, α).Thus any result of functions belonging to the classT(λ, α)aboutzF(a, b;c;z) leads to that of functions belonging to the class C(λ, α).Hence we obtain the following analogous result to TheoremA.

Theorem 3.1. Leta, b > −1, ab < 0andc > a+b+ 2.ThenG(a, b;c;z)defined by(3.1)is inC(λ, α)if and only if

c > a+b+ 1−(1−λα)ab 1−α .

Remark 2. Puttingλ= 0in the above results, we obtain the results of Silverman [7].

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References

[1] O. ALTINTASANDS. OWA, On subclasses of univalent functions with negative coefficients, Pusan Ky˘ongnam Math. J., 4 (1988), 41–56.

[2] B.C. CARLSONANDD.B. SHAFFER, Starlike and prestarlike hypergeometric functions, J. Math. Anal. Appl., 15 (1984), 737–745.

[3] N.E. CHO, S.Y. WOOANDS. OWA, Uniform convexity properties for hyperge- ometric functions, Fract. Calculus Appl. Anal., 5(3) (2002), 303–313.

[4] E. MERKESANDB.T. SCOTT, Starlike hypergeometric functions, Proc. Amer.

Math. Soc., 12 (1961), 885–888.

[5] St. RUSCHEWEYHANDV. SINGH, On the order of starlikeness of hypergeo- metric functions, J. Math. Anal. Appl., 113 (1986), 1–11.

[6] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer.

Math. Soc., 51 (1975), 109–116.

[7] H. SILVERMAN, Starlike and convexity properties for hypergeometric func- tions, J. Math. Anal. Appl., 172 (1993), 574–581.