Distortion and Convolutional Theorems for Operators of Generalized Fractional Calculus Involving Wright Function

Journal of Applied Analysis

Vol. 14, No. 2 (2008), pp. 183–192

DISTORTION AND CONVOLUTIONALTHEOREMS FOR OPERATORS OF

GENERALIZED FRACTIONAL CALCULUSINVOLVING WRIGHT FUNCTION

M. K. AOUF and J. DZIOK

Received July 9, 2007 and, in revised form, March 1, 2008

Abstract. Using the Wright’s generalized hypergeometric function, weinvestigate a class W (q, s;A,B, λ) of analytic functions with negativecoefficients. We derive many results for the modified Hadamard prod-uct of functions belonging to the class W (q, s;A,B, λ). Moreover, wegeneralize some of the distortion theorems to the classical fractionalintegrals and derivatives and the Saigo (hypergeometric) operators offractional calculus.

1. Introduction

Let A denote the class of functions f(z) of the form:

f(z) = z +∞∑k=2

akzk, (1)

which are analytic in the unit disc U = z : z ∈ C and |z| < 1.

2000 Mathematics Subject Classification. Primary: 30C45, 26A33.Key words and phrases. Wright’s generalized hypergeometric function, modified

Hadamard product, fractional calculus.

ISSN 1425-6908 c© Heldermann Verlag.

184 M. K. AOUF AND J. DZIOK

If f(z) and g(z) are analytic in U , we say that f(z) is subordinate tog(z), written symbolically f(z) ≺ g(z), if there exists a Schwarz functionw(z) (|w (z)| ≤ |z| in U), such that f(z) = g(w(z)) (z ∈ U).

For analytic functions

f(z) =∞∑k=0

akzk and g(z) =

∞∑k=0

bkzk,

by (f ∗ g)(z) we denote the Hadamard product (or convolution) of f(z) andg(z), defined by

(f ∗ g)(z) =∞∑k=0

akbkzk.

Let α1, A1, . . . , αq, Aq and β1, B1, . . . , βs, Bs (q, s ∈ N = 1, 2, . . . ) bepositive real parameters such that

1 +s∑

k=1

Bk −q∑

k=1

Ak ≥ 0.

The Wright generalized hypergeometric function [23] (see also [6], [12] and[19])

qΨs[(α1, A1), . . . , (αq, Aq); (β1, B1), . . . , (βs, Bs); z]

= qΨs[(αn, An)1,q; (βn, Bn)1,s; z]

is defined by

qΨs[(αk, Ak)1,q; (βk, Bk)1,s; z]

=∞∑k=0

q∏

n=1

Γ(αn + kAn)

s∏

n=1

Γ(βn + kBn)

−1zk

k!(z ∈ U). (2)

If An = 1 (n = 1, . . . , q) and Bn = 1 (n = 1, . . . , s), we have the relationship:

Ω qΨs[(αn, 1)1,q; (βn, 1)1,s; z] = qFs(α1, . . . , αq;β1, . . . , βs; z), (3)

where qFs(α1, . . . , αq;β1, . . . , βs; z) is the generalized hypergeometric func-tion (see for details the books on special functions, as [12], [19]) and

Ω =

(q∏

n=1

Γ(αn)

)−1( s∏n=1

Γ(βn)

). (4)

The Wright generalized hypergeometric functions (2) have been recentlyinvolved in the geometric function theory, see [1], [2], [3], [14], [15] and [16],as well as: [7], [8], [9] and [13]. It is a special case of Fox’s function (see forexample [6], [12] and [19]).

DISTORTION AND CONVOLUTIONAL THEOREMS 185

Using the Wright generalized hypergeometric functions Dziok and Raina[2] defined a function qφs[(αn, An)1,q; (βn, Bn)1,s; z] by

qφs[(αn, An)1,q; (βn, Bn)1,s; z] = ΩzqΨs[(αn, An)1,q; (βn, Bn)1,s; z]

and introduced the following linear operator

θ[(αn, An)1,q; (βn, Bn)1,s] : A → A,defined by the convolution

θ[(αn, An)1,q; (βn, Bn)1,s]f(z) = qφs[(αn, An)1,q; (βn, Bn)1,s; z] ∗ f(z).

We observe that for a function f(z) of the form (1) we have

θ[(αn, An)1,q; (βn, Bn)1,s]f(z) = z +∞∑k=2

Ωσkakzk, (5)

where Ω is given by (4) and σk is defined by

σk =Γ(α1 +A1(k − 1)) · · ·Γ(αq +Aq(k − 1)

Γ(β1 +B1(k − 1)) · · ·Γ(βs +Bs(k − 1))(k − 1)!. (6)

If, for convenience, we write

θ[α1]f(z) = θ[(α1, A1), . . . , (αq, Aq); (β1, B1), . . . , (βs, Bs)]f(z),

then one can easily verify from the definition (5) that

zA1(θ[α1]f(z))′

= α1θ[α1 + 1]f(z)− (α1 −A1)θ[α1]f(z). (7)

Using the linear operator θ[α1] Aouf and Dziok [1] defined the classW (q, s;A,B, λ) of functions of the form

f(z) = z −∞∑k=2

akzk, (ak ≥ 0) (8)

which also satisfy the following condition:1

(1− λ)

(α1θ[α1 + 1]f(z)θ[α1]f(z)

+A1(1− λ)− α1

)≺ A1

1 +Az

1 +Bz

(0 ≤ B ≤ 1;−B ≤ A < B; 0 ≤ λ < 1).

In particular, for q = s+1 and αs+1 = As+1 = 1, we write W (s;A,B, λ) =W (s+ 1, s;A,B, λ). The class W (q, s;A,B, 0) = W (q, s;A,B) was studiedby Dziok and Raina [2] (see also [3]).

If An = 1 (n = 1, . . . , q) and Bn = 1 (n = 1, . . . , s), then we notethat W (q, s;A,B, 0) = V 1

2 (q, s;A,B). This class was studied by Dziok andSrivastava [4] (see also [5]). Putting moreover α1 = n + 1, α2 = 1 andβ1 = 1, we have the class Tn(λ, ρ)=W (2, 1;−ρ, ρ, λ), which was studied byPatel and Acharya [11].

For the class W (q, s;A,B, λ) we have following result.


Lemma 1 ([1]). A function f(z) of the form (8) belongs to the classW (q, s;A,B, λ) if and only if

∞∑k=2

Ωδkak ≤ (B −A)(p− λ), (9)

where

δk = [(1 +B)(k − 1) + (B −A)(1− λ)]σk, (10)

and Ω, σk are defined by (4) and (6), respectively.

2. Modified Hadamard product

For the functions

fj(z) = z −∞∑k=2

ak,jzk (ak,j ≥ 0; j = 1, 2), (11)

we denote by (f1 ⊗ f2)(z) the modified Hadamard product or convolution ofthe functions f1 and f2 defined by

(f1 ⊗ f2)(z) = z −∞∑k=2

ak,1ak,2zk.

Theorem 1. Let the functions fj(z) (j = 1, 2) defined by (11) be in theclass W (q, s;A,B, λ). If the sequence δk is nondecreasing, then (f1 ⊗f2)(z) ∈W (q, s;A,B, γ), where

γ = 1− (1 +B)(B −A)(1− λ)2

Ω[(1 +B) + (B −A)(1− λ)]2σ2 − (B −A)2(1− λ)2 . (12)

The result is sharp.

Proof. We need to find the largest γ such that∞∑k=2

Ω[(1 +B)(k − 1) + (B −A)(1− γ)]σk(B −A)(1− γ)

ak,1ak,2 ≤ 1. (13)

By Lemma 1 and the Cauchy-Schwarz inequality, we obtain∞∑k=2

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]σk(B −A)(1− λ)

√ak,1ak,2 ≤ 1. (14)

Thus by (13) it is sufficient to show that

√ak,1ak,2 ≤

[(1 +B)(k − 1) + (B −A)(1− λ)](1− γ)[(1 +B)(k − 1) + (B −A)(1− γ)](1− λ)

(k ≥ 2).


By (14) we have

√ak,1ak,2 ≤

(B −A)(1− λ)Ω[(1 +B)(k − 1) + (B −A)(1− λ)]σk

(k ≥ 2).

Consequently, we need only to prove that

γ ≤ 1− (k − 1)(1 +B)(B −A)(1− λ)2

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]2σk − (B −A)2(1− λ)2

(k ≥ 2).

Since

Φ(k)

= 1− (k − 1)(1 +B)(B −A)(1− λ)2

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]2σk − (B −A)2(1− λ)2 (15)

is an increasing function of k (k ≥ 2), letting k = 2 in (15), we obtain

γ ≤ Φ(2) = 1 − (1 +B)(B −A)(1− λ)2

Ω[(1 +B) + (B −A)(1− λ)]2σ2 − (B −A)2(1− λ)2 ,

which proves the main assertion of Theorem 1. Finally, by taking the func-tions fj(z)(j = 1, 2) given by

fj(z) = z − (B −A)(1− λ)Ω[(1 +B) + (B −A)(1− λ)]σ2

z2 (j = 1, 2), (16)

we can see that the result is sharp.

Theorem 2. Let the functions fj(z) (j = 1, 2) defined by (11) be in theclass W (q, s;A,B, λ). If the sequence δk is nondecreasing, then the func-tion

h(z) = z −∞∑k=2

(a2k,1 + a2

k,2) zk (17)

belongs to the class W (q, s;A,B, τ), where

τ = 1− 2(1 +B)(B −A)(1− λ)2

Ω[(1 +B) + (B −A)(1− λ)]2σ2 − 2(B −A)2(1− λ)2 .

The result is sharp for the functions fj(z) (j = 1, 2) defined by (16).

Proof. By Lemma 1, we obtain∞∑k=2

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]σk

(B −A)(1− λ)

2

a2k,1


≤

∞∑k=2

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]σk(B −A)(1− λ)

ak,1

2

≤ 1 (18)

and∞∑k=2

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]σk

(B −A)(1− λ)

2

a2k,2

≤

∞∑k=2

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]σk(B −A)(1− λ)

ak,2

2

≤ 1. (19)

It follows from (18) and (19) that∞∑k=2

12

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]σk

(B −A)(1− λ)

2

(a2k,1 + a2

k,2) ≤ 1.

Therefore, we need to find the largest τ such thatΩ[(1 +B)(k − 1) + (B −A)(1− τ)]σk

(B −A)(1− τ)

≤ 12

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]σk

(B −A)(1− λ)

2

(k ≥ 2),

that is,

τ ≤ 1− 2(k − 1)(1 +B)(B −A)(1− λ)2

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]2σk − 2(B −A)2(1− λ)2

(k ≥ 2).

Since

D(k) = 1− 2(k − 1)(1 +B)(B −A)(1− λ)2

Ω[(1 +B)(k − 1) + (B −A)(1− λ)]2σk − 2(B −A)2(1− λ)2 ,

is an increasing function of k(k ≥ 2), we readily have

τ ≤ D(2) = 1− 2(1 +B)(B −A)(1− λ)2

Ω[(1 +B) + (B −A)(1− λ)]2σ2 − 2(B −A)2(1− λ)2 ,

and Theorem 2 follows at once.

Putting λ = 0 in Theorems 1 and 2, we obtain the following two corol-laries.

Corollary 1. Let the functions fj(z) (j = 1, 2) defined by (11) be in theclass W (q, s;A,B). If the sequence δk is nondecreasing, then (f1⊗f2)(z) ∈W (q, s;A,B, γ), where

γ = 1− (1 +B)(B −A)Ω(1 + 2B −A)2σ2 − (B −A)2 .



Corollary 2. Let the functions fj(z) (j = 1, 2) defined by (11) be in theclass W (q, s;A,B). If the sequence δk is nondecreasing, then the functionh(z) defined by (17) belongs to the class W (q, s;A,B, τ), where

τ = 1− 2(1 +B)(B −A)Ω(1 + 2B −A)2σ2 − 2(B −A)2 .


Taking q = 2, s = 1, A1 = A2 = B1 = 1, α1 = n + 1 ∈ N , α2 = β1 = 1,B = −A = ρ (0 < ρ ≤ 1) in Theorems 1 and 2, respectively, we obtain thefollowing consequences:

Corollary 3. Let the functions fj(z) (j = 1, 2) defined by (11) be in theclass Tn(λ, ρ). Then (f1 ⊗ f2)(z) ∈ Tn(γ, ρ), where

γ = 1− 2ρ(1 + ρ)(1− λ)2

[1 + ρ(3− 2λ)]2(n+ 1)− 4ρ2(1− λ)2 .


Corollary 4. Let the functions fj(z) (j = 1, 2) defined by (11) be in theclass Tn(λ, ρ). Then the function h(z) defined by (17) belongs to the classTn(τ, ρ), where

τ = 1− 4ρ(1 + ρ)(1− λ)2

[1 + ρ(3− 2λ)]2(n+ 1)− 8ρ2(1− λ)2 .


3. Definitions and applications of fractional calculus

We start with some definitions of fractional calculus operators (that isfractional derivatives and fractional integrals), as adopted for use in classesof analytic functions. First we recall the definition of Saigo operator [17],[18] (see also [21]).

For real numbers γ, ζ and η, the fractional derivative-integral operatorIγ,ζ,η0,z is defined by

Iγ,ζ,η0,z f(z) =z−γ−ζ

Γ(γ)

z∫0

(z − t)γ−12F1

(γ + ζ,−η; γ; 1− t

z

)f(t)dt for γ > 0

Iγ,ζ,η0,z f(z) =dn

dznIγ+n,ζ−η,η−n

0,z f(z) for 0 < γ + n ≤ 1, n = 1, 2, . . . ,


where f(z) is an analytic function in a simply connected region of thez-plane containing the origin with the order

f(z) = O(|z|ε) (z → 0),

where ε > max(0, ζ− η)− 1 and the multiplicity of (z− t)γ−1 is removed byrequiring log(z−t) to be real when z−t > 0. The function 2F1(a, b; c; z) is theGaussian hypergeometric function (3) defined, in terms of the Pochhammersymbol

(λ)n =

1 (n = 0)λ(λ+ 1) · · · (λ+ n− 1) (n ∈ N),

by

2F1(a, b; c; z) =∞∑n=0

(a)n(b)n(c)n

zn

n!(z ∈ U).

In particular we obtain the fractional derivative-integral operator

Dγz f(z) = Iγ,γ,η0,z f(z) (20)

introduced by Owa [10] (see also Srivastava and Owa [20]).In order to prove our result for the operator Iγ,ζ,η0,z , we recall here the

following lemma.

Lemma 2 ([22]). If n > max 0, ζ − η − 1, then

Iγ,ζ,η0,z zn =Γ(n+ 1)Γ(n− ζ + η + 1)

Γ(n− ζ + 1)Γ(n+ γ + η + 1)zn−ζ .

We can prove easy the following theorem.

Theorem 3. Let l ∈ N ∪ 0, −γ ≤ ζ ≤ l, ζ < 2 + η, and let thesequence δk/(k)l be nondecreasing. If the function f(z) is in the classW (q, s;A,B, λ) and 0 < |z| < 1, then∣∣∣Iγ,ζ,η0,z f(z)

∣∣∣ ≥ Γ(2− ζ + η)|z|1−ζ

Γ(2− ζ)Γ(2 + γ + η)

1− 2(B −A)(1− λ)(2− ζ + η)

Ωδ2(2− ζ)(2 + γ + η)|z|,∣∣∣Iγ,ζ,η0,z f(z)

∣∣∣ ≤ Γ(2− ζ + η)|z|1−ζ

Γ(2− ζ)Γ(2 + γ + η)

1 +

2(B −A)(1− λ)(2− ζ + η)Ωδ2(2− ζ)(2 + γ + η)

|z|.

The result is sharp for extremal function f of the form

f(z) = z − (B −A)(1− λ)Ωδ2

z2. (21)

Putting ζ = γ in Theorem 3 and using (20) we obtain the followingcorollary.


Corollary 5. Let l ∈ N ∪ 0, γ ≥ −l and let the sequence δk/(k)lbe nondecreasing. If the function f(z) is in the class W (q, s;A,B, λ) and0 < |z| < 1, then

|Dγz f(z)| ≥ |z|

1−γ

Γ(2− γ)

1− 2(B −A)(1− λ)

Ωδ2(2− γ)|z|,

|Dγz f(z)| ≤ |z|

1−γ

Γ(2− γ)

1 +

2(B −A)(1− λ)Ωδ2(2− γ)

|z|.

The result is sharp for extremal function f of the form (21).

Corollary 6. Let l ∈ N ∪ 0, γ ≥ −l and let the sequence δk/(k)l benondecreasing. If the function f(z) is in the class W (q, s;A,B, λ), thenDγz f(z) is included in a disc with its center at the origin and radius r given

by

r =1

Γ(2− γ)

1 +

2(B −A)(1− λ)Ωδ2(2− γ)

.

Remark 1. Taking λ = 0 in Theorem 3 and Corollaries 5 and 6, respec-tively, we obtain the results for the class W (q, s;A,B, 0) = W (q, s;A,B)studied by Dziok and Raina [2]. Corollaries 1 and 2 are improvements ofthe results obtained by Patel and Acharya [11].

Acknowledgements. The authors express their sincerest thanks to thereferees for useful comments.

References

[1] Aouf, M. K., Dziok, J., Certain class of analytic functions associated with the Wrightgeneralized hypergeometric function, J. Math. Appl. 30 (2008), 23–32.

[2] Dziok, J., Raina, R. K., Families of analytic functions associated with the Wrightgeneralized hypergeometric function, Demonstratio Math. 37(3) (2004), 533–542.

[3] Dziok, J., Raina, R. K., Srivastava, H. M., Some classes of analytic functions asso-ciated with operators on Hilbert space involving Wright’s generalized hypergeometricfunction, Proc. Jangieon Math. Soc. 7 (2004), 43–55.

[4] Dziok, J., Srivastava, H. M., Certain subclasses of analytic functions associated withthe generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (2003),7–18.

[5] Dziok, J., Srivastava, H. M., Classes of analytic functions associated with the gener-alized hypergeometric function, Appl. Math. Comput. 103 (1999), 1–13.

[6] Kiryakova, V., Generalized Fractional Calculus and Applications, Longman & J. Wi-ley, Harlow-New York, 1994.

[7] Kiryakova, V., Saigo, M., Criteria for generalized fractional integrals to preserveunivalency of analytic functions, C. R. Acad. Bulgare Sci. 58(10) (2005), 1127–1134.


[8] Kiryakova, V., Saigo, M., Owa, S., Distortion and characterization theorems forgeneralized fractional integration operators involving H-function in subclasses of uni-valent functions, Fukuoka Univ. Sci. Rep. 34(1) (2004), 1–16.

[9] Kiryakova, V., Saigo, M., Srivastava, H. M., Some criteria for univalence of analyticfunctions involving generalized fractional calculus operators, Fract. Calc. Appl. Anal.1(1) (1998), 79–104.

[10] Owa, S., On the distortion theorems. I, Kyungpook Math. J. 18 (1978), 53–59.[11] Patel, J., Acharya, M., Certain subclasses of starlike functions with negative coeffi-

cients, Bull. Calcuta Math. Soc. 87 (1995), 265–276.[12] Prudnikov, A. P., Brychkov, Yu. A., Marichev, O. I., Integrals and Series. Vol. 3:

More Special Functions, Gordon & Breach Sci. Publ., New York, 1990.[13] Raina, R. K., Kalia, R. N., Characterizations for subclasses of analytic functions

connecting linear fractional calculus operators, Fract. Calc. Appl. Anal. 1(4) (1998),335–350.

[14] Raina, R. K., Nahar, T. S., A note on boundedness properties of Wright’s generalizedhypergeometric function, Ann. Math. Blaise Pascal 4 (1997), 83–95.

[15] Raina, R. K., Nahar, T. S., On characterization of certain Wright’s generalized hy-pergeometric functions involving certain subclasses of analytic functions, Informatica(Vilnius) 10 (1999), 219–230.

[16] Raina, R. K., Nahar, T. S., On univalent and starlike Wright’s hypergeometric func-tions, Rend. Sem. Mat. Univ. Padova 95 (1996), 11–22.

[17] Saigo, M., A certain boundary value problem for the Euler-Darboux equation, Math.Japon. 24 (1980), 377–385.

[18] Saigo, M., A remark on integral operators involving the Gauss hypergeometric func-tions, Math. Rep. Kyushu Univ. 11 (1978), 135–143.

[19] Srivastava, H. M., Gupta, C. K., Goyal, S. P., The H-Functions of One and TwoVariables with Applications, South Asian Publ., New Delhi, 1982.

[20] Srivastava, H. M., Owa, S., An application of the fractional derivative, Math. Japon.29 (1984), 383–389.

[21] Srivastava, H. M., Owa, S., (editors), Current Topics in Analytic Function Theory,World Sci. Publ. Comp., Singapore-New Jersey-London-Hong Kong, 1992.

[22] Srivastava, H. M., Saigo, M., Owa, S., A class of distortion theorems involving certainoperators of fractional calculus, J. Math. Anal. Appl. 131 (1988), 412–420.

[23] Wright, E. M., The asymptotic expansion of the generalized hypergeometric function,Proc. London Math. Soc. (3) 46 (1946), 389–408.

Mohamed Kamal Aouf Jacek Dziok

Department of Mathematics Institute of Mathematics

Faculty of Science University of Rzeszow

Mansoura University Rejtana 16A

Mansoura 35516, Egypt PL-35-310 Rzeszow, Poland

e-mail: [email protected] e-mail: [email protected]

Documents

Distortion and Convolutional Theorems for Operators of Generalized Fractional Calculus Involving Wright Function