Volume 10 (2009), Issue 1, Article 14, 7 pp.
SOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS
INVOLVING APPELL HYPERGEOMETRIC FUNCTION
R. K. RAINA
10/11 G ANPATI V IHAR , O PPOSITE S ECTOR 5
U DAIPUR 313002, R AJASTHAN , I NDIA
rkraina_7@hotmail.com
Received 14 August, 2008; accepted 11 January, 2009
Communicated by H.M. Srivastava
A BSTRACT. A class of fractional derivative operators (with the Appell hypergeometric function
in the kernel) is used here to define a new subclass of analytic functions and a coefficient bound
inequality is established for this class of functions. Also, an inclusion theorem for a class of
fractional integral operators involving the Hardy space of analytic functions is proved. The
concluding remarks briefly mentions the relevances of the main results and possibilities of further
work by using these new classes of fractional calculus operators.
Key words and phrases: Analytic functions, Hardy space, Fractional derivatives and fractional integrals, Appell hypergeometric function, Inclusion relation.
2000 Mathematics Subject Classification. 26A33, 30C45.
1. I NTRODUCTION , D EFINITIONS AND P RELIMINARIES
Let A(n) denote the class of functions f (z) normalized by
f (z) = z +
(1.1)
∞
X
ak z k
(n ∈ N),
and
|z| < 1}.
k=n+1
which are analytic in the open unit disk
U = {z : z ∈ C
(α,α0 ,β,β 0 ,γ)
We denote by ∆n
(σ) the subclass of functions in A(n) which also satisfy the inequality:
n
o
0
0
0
0
α+α0 +γ−1 (α,α ,β,β ,γ)
(1.2)
Re χ1 (α, α , β, β , γ)z
D0,z
f (z) > σ
(z ∈ U),
230-08
2
R. K. R AINA
(α,α0 ,β,β 0 ,γ)
where D0,z
convenience)
is the generalized fractional derivative operator (defined below), and (for
(1.3) χm (α, α0 , β, β 0 , γ)
=
Γ(1 + m + β 0 )Γ(1 + m − α − α0 − γ)Γ(1 + m − α0 − β − γ)
Γ(1 + m)Γ(1 + m − α0 + β 0 )Γ(1 + m − α − α0 − β − γ)
(m ∈ N),
provided that
0 ≤ σ < 1; 0 ≤ γ < 1;
(1.4)
γ< min (−α − α0 , −α0 − β, −α − α0 − β) + m + 1;
β 0 > max(0, α0 ) − m − 1.
Following [8], a function f (z) is said to be in the class Vn (θk ) if f (z) ∈ A(n) satisfies the
condition that
arg(ak ) = θk (k ≥ n + 1; n ∈ N)
and if there exists a real number ρ such that
θk + (k − 1)ρ ≡ π(mod2π)
(1.5)
(k ≥ n + 1; n ∈ N),
then we say that f (z) is in the class Vn (θk ; ρ). Suppose Vn = ∪Vn (θk ; ρ) over all possible
(α,α0 ,β,β 0 ,γ)
sequences θk with ρ satisfying (1.5), then we denote by ∇n
(σ) the subclass of Vn
(α,α0 ,β,β 0 ,γ)
which consists of functions f (z) belonging to the class ∆n
(σ).
We present here the following family of fractional integral (and derivative) operators which
involve the familiar Appell hypergeometric function F3 (see also Kiryakova [4] and Saigo and
Maeda [9]).
(α,α0 ,β,β 0 ,γ)
Definition 1.1. Let γ > 0 and α, α0 , β, β 0 ∈ R. Then the fractional integral operator I0,z
of a function f (z) is defined by
(α,α0 ,β,β 0 ,γ)
(1.6) I0,z
f (z)
−α Z
z
=
Γ(γ)
z
γ−1 −α0
(z − ζ)
ζ
F3
0
ζ
z
α, α , β, β ; γ; 1 − , 1 −
z
ζ
0
0
f (ζ)dζ
(γ> 0),
where the function f (z) is analytic in a simply-connected region of the complex z-plane containing the origin, and it is understood that (z − ζ)γ−1 denotes the principal value for 0 5
arg(z − t) < 2π. The function F3 occurring in the kernel of (1.6) is the familiar Appell hypergeometric function of third type (also known as Horn’s F3 - function; see, for example, [10])
defined by
∞ X
∞
X
(α)m (α0 )n (β)m (β 0 )n z m ξ n
0
0
(1.7) F3 (α, α , β, β ; γ; z, ξ) =
(|z| < 1, |ξ| < 1) ,
(γ)m+n
m! n!
m=0 n=0
which is related to the Gaussian hypergeometric function 2 F1 (α, β; γ; z) by the following relationship:
2 F1 (α, β; γ; z)
= F3 (α, α0 , β, β 0 ; γ; z, 0)
= F3 (α, 0, β, β 0 ; γ; z, ξ) = F3 (α, α0 , β, 0; γ; z, ξ) .
(α,α0 ,β,β 0 ,γ)
Definition 1.2. The fractional derivative operator D0,z
(1.8)
(α,α0 ,β,β 0 ,γ)
D0,z
f (z) =
dn (α,α0 ,β−n,β 0 ,n−γ)
I
f (z)
dz n 0,z
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 14, 7 pp.
of a function f (z) is defined by
(n − 1 ≤ γ < n; n ∈ N).
http://jipam.vu.edu.au/
S OME R ESULTS A SSOCIATED W ITH F RACTIONAL C ALCULUS O PERATORS I NVOLVING A PPELL H YPERGEOMETRIC F UNCTION
3
It may be observed that for
α0 = β 0 = 0,
α = λ + µ,
(1.9)
β = −η,
γ = λ,
we obtain the relationship
(λ+µ,0,−η,0,λ)
I0,z
(1.10)
λ,µ,η
= I0,z
λ,µ,η
in terms of the Saigo type fractional integral operator I0,z
([12]). On the other hand, if
α0 = β 0 = 0,
α = µ − λ,
(1.11)
β = −η,
γ = λ,
then we get
(µ−λ,0,−η,0,λ)
D0,z
(1.12)
(λ,µ,η
where J0,z
(1.13)
λ,µ,η
,
= J0,z
is the Saigo type fractional derivative operator ([6]; see also [7]). Further, when
α = β 0 = 0,
(0,1−µ,0,0,λ)
α0 = 1 − µ,
γ = λ (or − λ),
(0,1−µ,0,0,−λ)
then the operators I0,z
and D0,z
correspond to the differential-integral operaλ
tors Qµ due to Dziok [2].
Let Hp (0 ≤ p < ∞) be the class of analytic functions in U such that
kf kp = lim {Mp (r, f )} < ∞,
(1.14)
r→1−
where
(1.15)
kf kp =
 R 1
2π 1

iθ p p
 2π
f
(re
)
(0 < p < ∞),
0

 sup |f (z)| .
|z|≤r
In this paper we first define a new function class in terms of the fractional derivative operators
(with the Appell hypergeometric function in the kernel) and then establish a coefficient bound
inequality for this function class. Also, we prove an inclusion theorem for a class of fractional
integral operators involving the Hardy space of analytic functions. The relevance of the main
results and possibilities of further work by using the new classes of fractional calculus operators
are briefly pointed out in the concluding section of this paper.
2. A S ET OF C OEFFICIENT B OUNDS
We begin by proving the following coefficient bounds inequality for a function f (z) to be in
(α,α0 ,β,β 0 ,γ)
the class ∆n
(σ).
(α,α0 ,β,β 0 ,γ)
Theorem 2.1. Let f (z) defined by (1.1) be in the class ∆n
(2.1)
(σ), then
∞
X
|ak |
1−σ
≤
,
0
0
0
0
χ
χ
k (α, α , β, β , γ)
1 (α, α , β, β , γ)
k=n+1
where χm (α, α0 , β, β 0 , γ) is defined by (1.3). The result is sharp.
Proof. Assume that
n
o
0
0
0
0
α+α0 −γ−1 (α,α ,β,β ,γ)
Re χ1 (α, α , β, β , γ)z
D0,z
f (z) > σ
(z ∈ U).
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 14, 7 pp.
http://jipam.vu.edu.au/
4
R. K. R AINA
Using (1.1) and the formula (see, e.g. [9, p. 394]):
(2.2)
(α,α0 ,β,β 0 ,γ) q
D0,z
z
=
Γ(1 + q)Γ(1 + q − α0 + β 0 )Γ(1 + q − α − β − γ)
0
z q−α−α −γ ,
0
0
0
Γ(1 + q + β )Γ(1 + q − α − β − γ)Γ(1 + q − α − α − γ)
(0 ≤ γ < 1; α, α0 , β, β 0 ∈ R; q > max (0, α0 − β 0 , α + β + γ) − 1)
we obtain
(
∞
X
χ1 (α, α0 , β, β 0 , γ)
Re 1 +
ak z k−1
0
0
χ
k (α, α , β, β , γ)
k=n+1
(2.3)
)
>σ
(z ∈ U),
and for f (z) ∈ Vn (θk ; ρ) (z = reiθ ), the inequality thus obtainable from (2.3) on letting r → 1−
therein, readily yields
(
)
∞
X
χ1 (α, α0 , β, β 0 , γ)
(2.4)
Re 1 +
|ak | exp (i (θk + (k − 1) ρ)) > σ.
χk (α, α0 , β, β 0 , γ)
k=n+1
If we apply (1.5), then (2.4) gives
1−
(2.5)
∞
X
χ1 (α, α0 , β, β 0 , γ)
|ak | > σ,
0
0
χ
k (α, α , β, β , γ)
k=n+1
which leads to the desired inequality (2.1). We also observe that the equality sign in (2.1) is
attained for the function f (z) defined by
f (z) = z +
(2.6)
(1 − σ)χk (α, α0 , β, β 0 , γ) k
z exp (iθk )
χ1 (α, α0 , β, β 0 , γ)
(k ≥ n + 1; n ∈ N),
and this completes the proof of Theorem 2.1.
3. I NCLUSION R ELATIONS
Under the hypotheses of Definition 1.1, let
γ > 0; min (γ − α − α0 , γ − α0 − β, β 0 , γ − α − α0 − β, β 0 − α0 ) > −2;
(3.1)
α, α0 , β, β 0 ∈ R,
then the fractional integral operator
0
0
Ωz(α,α ,β,β ,γ) : A → A
(A(1) = A)
is defined by
0
0
0
(α,α0 ,β,β 0 ,γ)
,β,β ,γ)
Ω(α,α
f (z) = χ1 (α, α0 , β, β 0 , −γ)z α+α +γ I(0,z)
z
(3.2)
f (z).
where χ1 (α, α0 , β, β 0 , −γ) is given by (1.3).
By using the formula ([9, p. 394]; see also [4, p. 170, Lemma 9])
(α,α0 ,β,β 0 ,γ) q
(3.3) I0,z
z
=
Γ(1 + q)Γ(1 + q − α0 + β 0 )Γ(1 + q − α − α0 − β + γ) q−α−α0 +γ
z
,
Γ(1 + q + β 0 )Γ(1 + q − α0 − β + γ)Γ(1 + q − α − α0 + γ)
(γ > 0; α, α0 , β, β 0 ∈ R; q > max (0, α0 − β 0 , α + β − γ) − 1)
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 14, 7 pp.
http://jipam.vu.edu.au/
S OME R ESULTS A SSOCIATED W ITH F RACTIONAL C ALCULUS O PERATORS I NVOLVING A PPELL H YPERGEOMETRIC F UNCTION
5
it follows from (1.1), (3.2) and (3.3) that
(3.4)
0 ,β,β 0 ,γ)
Ω(α,α
f (z)
z
0
0
= z + χ1 (α, α , β, β , −γ)
∞
X
k=2
χk
ak
zk ,
0
(α, α , β, β 0 , −γ)
where (as before) χk (α, α0 , β, β 0 , −γ) is given by (1.3).
Before stating and proving our main inclusion theorem, we recall here the following known
results concerning the class R(ρ) in A which satisfies the inequality that <{f 0 (z)} > ρ(0 ≤
ρ < 1), where R(1) is denoted by R.
Lemma 3.1 ([3, p. 141]). Let f (z) ∈ R, then
f (z) ∈ Hp
(3.5)
: (0 < p < ∞).
Lemma 3.2 ([5, p. 533]). Let f (z) defined by (1.1) be in the class R(ρ) (0 ≤ ρ < 1), then
2
(k = 2, 3, 4, ...).
(3.6)
|ak | ≤
k
Theorem 3.3. Let f (z) ∈ R, then (under the constraints stated in (3.1))
0
0
,β,β ,γ)
Ω(α,α
f (z) ∈ Hp
z
(3.7)
(0 < p < ∞)
and
0
0
,β,β ,γ)
Ω(α,α
f (z) ∈ H∞
z
(3.8)
(γ > 1).
Proof. In view of (1.6) and (3.2), we obtain
0
0
,β,β ,γ)
(3.9) Ω(α,α
f (z) = χ1 (α, α0 , β, β 0 , −γ)
z
Z 1
1
γ−1 −α0
0
0
×
(1 − t) t F3 α, α , β, β ; γ; 1 − t, 1 −
f (zt)dt.
t
0
This implies that
d (α,α0 ,β,β 0 ,γ)
Ω
f (z)
(3.10) Re
dz z
Z 1
1
0
0
γ−1 1−α0
0
0
= χ1 (α, α , β, β , −γ)
(1 − t) t
F3 α, α , β, β ; γ; 1 − t, 1 −
< {f 0 (zt)} dt.
t
0
Since f (z) ∈ R, therefore, we infer from (3.10) that
0
0
Ωz(α,α ,β,β ,γ) f (z) ∈ R,
(3.11)
and applying Lemma 3.1, (3.11) gives the inclusion relation (3.7) under the conditions stated in
(3.1).
To prove the result (3.8), we observe the following three-term recurrence relation:
(3.12)
d (α,α0 ,β,β 0 ,γ)
Ω
f (z)
dz z
n
o
0 ,β,β 0 ,γ−1)
0
(α,α0 ,β,β 0 ,γ)
= z −1 (γ − α0 − β + 1) Ω(α,α
f
(z)
−
(γ
−
α
−
β)
Ω
f
(z)
,
z
z
which yields the inequality
p
p
n
d (α,α0 ,β,β 0 ,γ)
0 ,β,β 0 ,γ−1)
p
(3.13) Ωz
f (z) ≤ r−p (γ − α0 − β + 1) Ω(α,α
f
(z)
z
dz
p o
p (α,α0 ,β,β 0 ,γ)
0
− (γ − α − β) Ωz
f (z)
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 14, 7 pp.
(|z| = r) ,
http://jipam.vu.edu.au/
6
R. K. R AINA
provided that
(3.14)
γ > 1; min (1+γ −α−α0 , 1+γ −α0 −β, 1+β 0 , 1+γ −α−α0 −β, 1+β 0 −α0 ) > −1;
α, α0 , β, β 0 ∈ R
and 0 < p < ∞.
Making use of (1.14) and (1.15), the above inequality (3.13) (with p = 1) yields
n
d (α,α0 ,β,β 0 ,γ)
0 ,β,β 0 ,γ−1)
(3.15) M1 r, Ωz
f (z) ≤ r−1 (γ − α0 − β + 1) M1 r, Ω(α,α
f
(z)
z
dz
o
0
(α,α0 ,β,β 0 ,γ)
− (γ − α − β) M1 r, Ωz
f (z)
and
(3.16)
d (α,α0 ,β,β 0 ,γ)
(α,α0 ,β,β 0 ,γ−1)
Ωz
≤ (γ − α0 − β + 1) f
(z)
Ω
f
(z)
z
dz
1
1
(α,α0 ,β,β 0 ,γ)
0
− (γ − α − β) Ωz
f (z) .
1
Applying (3.7), we infer (under the constraints stated in (3.14)) that
(3.17)
0
0
,β,β ,γ−1)
Ω(α,α
f (z) ∈ H1
z
0
0
,β,β ,γ)
and Ω(α,α
f (z) ∈ H1
z
(γ > 1),
and consequently (3.16) implies that
d (α,α0 ,β,β 0 ,γ)
Ω
f (z) ∈ H1 ,
dz z
provided that the conditions stated in (3.14) are satisfied. By appealing to a known result
(α,α0 ,β,β 0 ,γ)
[1, p. 42, Theorem 3.11], we infer from (3.17) that Ωz
f (z) is continuous in U∗ =
(α,α0 ,β,β 0 ,γ)
{z : z ∈ C and |z| ≥ 1} . But U∗ being compact, we finally conclude that Ωz
f (z) is a
bounded analytic function in U, and the proof of the second assertion (3.8) of Theorem 3.3 is
complete.
The assertion (3.8) of Theorem 3.3 can also be proved by applying Lemma 3.2 (see also [3,
p. 145]). Indeed, it follows from (3.4) and (3.6) that
∞
X
k
|ak |
(α,α0 ,β,β 0 ,γ)
0
0
z f (z) ≤ |z| + χ1 (α, α , β, β , −γ)
Ωz
0
0
χ
(α,
α
,
β,
β
,
−γ)
k
k=2
0
0
0
0
≤ 1 + 2 : χ1 (α, α , β, β , −γ)
∞
X
k=2
=1+
Γ(k)
Γ(k + 1)χk (α, α0 , β, β 0 , −γ)
2(2 − α + β )(2 − α − α0 − β + γ)
(2 + β 0 )(2 − α − α0 + γ)(2 − α0 − β + γ)
1, 2, 3 − α0 + β 0 , 3 − α − α0 − β + γ;
× 4 F3
3 + β 0 , 3 − α − α0 + γ, 3 − α0 − β + γ;
1
in terms of the generalized hypergeometric function.
Now, for fixed values of the parameters α, α0 , β, β 0 , γ satisfying the conditions stated in (3.1),
we observe that by using the asymptotic formula [10, p. 109],
Γ(k)
−γ−1
=
o
k
(k → ∞),
Γ(k + 1)χk (α, α0 , β, β 0 , −γ)
and since γ > 1, this proves our assertion (3.8).
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 14, 7 pp.
http://jipam.vu.edu.au/
S OME R ESULTS A SSOCIATED W ITH F RACTIONAL C ALCULUS O PERATORS I NVOLVING A PPELL H YPERGEOMETRIC F UNCTION
7
4. C ONCLUDING R EMARKS
In view of the relationships (1.10) and (1.12), the main results (Theorems 2.1 and 3.3) of this
paper would correspond to the results due to Raina and Srivastava [8, p. 75, Theorem 1; p. 79,
Theorem 7]. Furthermore, in view of the relationship (1.13), we can easily apply Theorems
2.1 and 3.3 to obtain the corresponding results associated with Dziok’s differential-integral
operators [2]. The family of fractional calculus operators (fractional integrals and fractional
derivatives) defined by (1.6) and (1.8) can fruitfully be used in Geometric Function Theory.
Several new analytic, multivalent (or meromorphic) function classes can be defined and the
various properties of coefficient estimates, distortion bounds, radii of starlikeness, convexity
and close to convexity for such contemplated classes investigated.
R EFERENCES
[1] P.L. DUREN, Theory of H p Spaces, Vol. 38, A series of monographs and textbooks in pure and
applied mathematics, Academic Press, New York, 1970.
[2] J. DZIOK, Applications of the Jack lemma, Acta Math. Hungar., 105 (2004), 93–102.
[3] I. B. JUNG, Y. C. KIM AND H. M. SRIVASTAVA, The Hardy space of analytic functions associated
with certain one-parameter families of integral operators, J. Math. Anal. Appl., 176 (1993), 138–
147.
[4] V. KIRYAKOVA, On two Saigo’s fractional integal operators in the class of univalent functions,
Fracl. Cal. Appld. Math., 9 (2006), 159–176.
[5] T.H. MACGREGOR, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc.,
104 (1962), 532–537.
[6] R.K. RAINA AND T.S. NAHAR, Characterization properties for starlikeness and convexity of some
subclasses of analytic functions involving a class of fractional derivative operators, Acta Math.
Univ. Comenianae, 69 (2000), 1–8.
[7] R.K. RAINA AND H.M. SRIVASTAVA, A certain subclass of analytic functions associated with
operators of fractional calculus, Comput. Math. Appl., 32 (1996), 13–19.
[8] R.K. RAINA AND H.M. SRIVASTAVA, Some subclasses of analytic functions associated with
fractional calculus operators, Comput. Math. Appl., 37 (1999), 73–84.
[9] M. SAIGO AND N. MAEDA, More generalization of fractional calculus, In:Transform Methods
and Special Functions, Varna’96 (Proc. Second Internat. Workshop), Science Culture Technology
Publishing, Singapore (1998), 386-400.
[10] H.M. SRIVASTAVA AND P.W. KARLSSON, Multiple Gaussian Hypergeometric Series, Halsted
Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1985.
[11] H.M. SRIVASTAVA AND S. OWA, Current Topics in Analytic Function Theory, World Scientific
Publishing Company, Singapore, New Jersey, London and Hongkong, 1992.
[12] H.M. SRIVASTAVA, M. SAIGO AND S. OWA, A class of distortion theorems involving certain
operators of fractional calculus, J. Math. Anal. Appl., 131 (1988), 412–420.
J. Inequal. Pure and Appl. Math., 10(1) (2009), Art. 14, 7 pp.
http://jipam.vu.edu.au/