Paper-12-Acta23.2010. 115KB Jun 04 2011 12:03:10 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 23/2010
pp. 123-132
SOME APPLICATIONS OF FRACTIONAL CALCULUS
OPERATORS TO CERTAIN SUBCLASS OF ANALYTIC
FUNCTIONS WITH NEGATIVE COEFFICIENTS
B.A. Frasin
Abstract. The object of the present paper is to derive various distortion theorems for fractional calculus and fractional integral operators of functions in the class
BT(j, λ, α) consisting of analytic and univalent functions with negative coefficients.
Furthermore, some of integral operators of functions in the class BT(j, λ, α) is shown.
2000 Mathematics Subject Classification: 30C45.
1.Introduction and definitions
Let A(j) denote the family of functions of the form:
f (z) = z +
∞
X
an z n
(j ∈ N = {1, 2, 3, · · · }),
(1)
n=j+1
which are analytic in the open unit disk U = {z : |z| < 1}. A function f (z)
belonging to A(j) is in the class B(j, λ, α) if and only if
zf ′ (z) + (2λ2 − λ)z 2 f ′′ (z)
>α
(2)
Re
4(λ − λ2 )z + (2λ2 − λ)zf ′ (z) + (2λ2 − 3λ + 1)f (z)
for some α(0 ≤ α < 1) and λ(0 ≤ λ < 1), and for all z ∈ U.
Let T(j) denote the subclass of A(j) consisting of functions of the form:
f (z) = z −
∞
X
an z n
(an ≥ 0, j ∈ N),
(3)
n=j+1
Further, we define the class BT(j, λ, α) by
BT(j, λ, α) = B(j, λ, α) ∩ T(j).
123
(4)
B.A. Frasin - Some applications of fractional calculus operators...
The class BT(j, λ, α) was introduced and studied by the author in [3]. The
class BT(j, λ, α) is of special interest because it reduces to various classes of wellknown functions as well as many new ones. For example The classes BT(1, 0, α) =
T ∗ (α) and BT(1, 1, α) = C(α) were first studied by Silverman [10]. The classes
BT(j, 0, α) = Tα∗ (j) and BT(j, 1, α) = Cα (j) were studied Srivastava et al. [13]. The
class BT(1, 1/2, α) = BT(α) was studied by Gupta and Jain [4].
In order to show our results, we shall need the following lemma.
Lemma 1. ([3]) Let the function f (z) be defined by (3). Then f (z) ∈ BT(j, λ, α)
if and only if
∞
X
σ(n, α, λ)an ≤ 1 − α,
(5)
n=j+1
where
σ(n, α, λ) := (2λ2 − λ)n2 + [1 + (1 + α)(λ − 2λ2 )]n + (1 + 2λ2 − 3λ)α
(6)
and 0 ≤ α < 1, 0 ≤ λ < 1. The result is sharp.
2.Fractional calculus
Many essentially equivalent definitions of fractional calculus (that is fractional
derivatives and fractional integrals) have been given in the literature (cf., e.g., [1],
[2, Chap. 13], [5],[7], [8], [9], [11, p.28 et. seq.]. We find it to be convenient to recall
here the following definitions which are used earlier by Owa [6] (and, subsequently,
by Srivastava and Owa [12]).
Definition 1. The fractional integral of order µ is defined, for a function f (z),
by
Dz−µ f (z)
1
=
Γ(µ)
Zz
f (ζ)
dζ,
(z − ζ)1−µ
(7)
0
where µ > 0, f (z) is an analytic function in a simply-connected region of the z-plane
containing the origin, and the multiplicity of (z − ζ)1−µ is removed by requiring log
(z − ζ) to be real when z − ζ > 0.
Definition 2. The fractional derivative of order µ is defined, for a function
f (z), by
Dzµ f (z)
d
1
=
Γ(1 − µ) dz
Zz
0
124
f (ζ)
dζ,
(z − ζ)µ
(8)
B.A. Frasin - Some applications of fractional calculus operators...
where 0 ≤ µ < 1, f (z) is an analytic function in a simply-connected region of the
z-plane containing the origin, and the multiplicity of (z − ζ)−µ is removed as in
Definition 1 above.
Definition 3. Under the hypotheses of Definition 2, the fractional derivative of
order n + µ is defined by
dn µ
D f (z),
dz n z
where 0 ≤ µ < 1 and n ∈ N0 = {0, 1, 2, . . .}.
We begin by proving
Dzn+µ f (z) =
(9)
Theorem 1. If f (z) ∈ BT (j, λ, α), then
and
1+µ
−µ
Dz f (z) ≥ |z|
Γ(2 + µ)
1+µ
−µ
Dz f (z) ≤ |z|
Γ(2 + µ)
(10)
(11)
(1 − α)Γ(j + 2)Γ(2 + µ)
|z|j
1−
σ(j + 1, α, λ)Γ(j + 2 + µ)
(1 − α)Γ(j + 2)Γ(2 + µ)
|z|j
σ(j + 1, α, λ)Γ(j + 2 + µ)
1+
,
for µ > 0 and z ∈ U. The results (10) and (11) are sharp.
Proof . Define the function G(z)by
G(z) = Γ(2 + µ)z −µ Dz−µ f (z)
∞
X
Γ(n + 1)Γ(2 + µ)
= z−
an z n
Γ(n + 1 + µ)
= z−
n=j+1
∞
X
Ψ(n)an z n ,
n=j+1
where
Ψ(n) =
Γ(n + 1)Γ(2 + µ)
Γ(n + 1 + µ)
(n ≥ j + 1).
(12)
It easy to see that
0 < Ψ(n) ≤ Ψ(j + 1) =
125
Γ(j + 2)Γ(2 + µ)
.
Γ(j + 2 + µ)
(13)
B.A. Frasin - Some applications of fractional calculus operators...
Furthermore, it follows from Lemma 1 that
∞
X
an ≤
n=j+1
1−α
,
σ(j + 1, α, λ)
(14)
Therefore, by using (13) and (14), we can see that
|G(z)| ≥ |z| − Ψ(j + 1) |z|j+1
∞
X
an ≥ |z| −
n=j+1
(1 − α)Γ(j + 2)Γ(2 + µ)
|z|j+1 (15)
σ(j + 1, α, λ)Γ(j + 2 + µ)
and
|G(z)| ≤ |z|+Ψ(j +1) |z|j+1
∞
X
an ≤ |z|+
n=j+1
(1 − α)Γ(j + 2)Γ(2 + µ)
|z|j+1 , (16)
σ(j + 1, α, λ)Γ(j + 2 + µ)
which prove the inequalities of Theorem 1.
Finally, we can easily see that the results (10) and (11) are sharp for the function
f (z) given by
(1 − α)Γ(j + 2)Γ(2 + µ) j
z 1+µ
−µ
(17)
1−
z
Dz f (z) =
Γ(2 + µ)
σ(j + 1, α, λ)Γ(j + 2 + µ)
or
f (z) = z −
1−α
z j+1 .
σ(j + 1, α, λ)
(18)
Theorem 2. If f (z) ∈ BT (j, λ, α), then
|Dzµ f (z)|
|z|1−µ
≥
Γ(2 − µ)
|Dzµ f (z)|
|z|1−µ
≤
Γ(2 − µ)
and
(19)
(20)
(1 − α)Γ(j + 2)Γ(2 − µ)
|z|j
1−
σ(j + 1, α, λ)Γ(j + 2 − µ)
(1 − α)Γ(j + 2)Γ(2 − µ)
|z|j
1+
σ(j + 1, α, λ)Γ(j + 2 − µ)
for 0 ≤ µ < 1 and z ∈ U. The results (19) and (20) are sharp.
Proof. Define the function H(z) by
126
,
B.A. Frasin - Some applications of fractional calculus operators...
H(z) = Γ(2 − µ)z µ Dzµ f (z)
∞
X
Γ(n + 1)Γ(2 − µ)
= z−
an z n
Γ(n + 1 − µ)
= z−
n=j+1
∞
X
Φ(n)an z n ,
n=j+1
where
Γ(n)Γ(2 − µ)
Γ(n + 1 − µ)
Φ(n) =
(n ≥ j + 1).
(21)
Γ(j + 1)Γ(2 − µ)
.
Γ(j + 2 − µ)
(22)
It easy to see that
0 < Φ(n) ≤ Φ(j + 1) =
Consequently, with the aid of (14) and (22), we have
|H(z)| ≥ |z|−Φ(j +1) |z|j+1
∞
X
nan ≥ |z|−
n=j+1
(1 − α)Γ(j + 2)Γ(2 − µ)
|z|j+1 (23)
σ(j + 1, α, λ)Γ(j + 2 − µ)
and
j+1
|H(z)| ≤ |z| + Φ(j + 1) |z|
∞
X
nan ≤ |z| +
n=j+1
(1 − α)Γ(j + 2)Γ(2 − µ)
|z|j+1 .
σ(j + 1, α, λ)Γ(j + 2 − µ)
(24)
Now (19) and (20) follow from (23) and (24), respectively.
Finally, by taking the function f (z) defined by
(1 − α)Γ(j + 2)Γ(2 − µ) j
z 1−µ
1−
z
Dzµ f (z) =
Γ(2 − µ)
σ(j + 1, α, λ)Γ(j + 2 − µ)
(25)
or for the function given by (18), the results (19) and (20) are easily seen to be
sharp.
Remark 1. Letting µ = 0 in Theorem 1 and µ −→ 1 in Theorem 2, we shall
obtain the corresponding results Theorem 3 and Theorem 4 in [3].
127
B.A. Frasin - Some applications of fractional calculus operators...
3.Fractional integral operator
We need the following definition of fractional integral operator given by Srivastava et al. [14].
Definition 4. For real number η > 0, γ and δ, the fractional integral operator
η,γ,δ
I0,z
is defined by
η,γ,δ
I0,z
f (z)
z −η−γ
=
Γ(η)
Zz
(z − t)η−1 F (η + γ, −δ; η; 1 − t/z)f (t)dt,
(26)
0
where a function f (z) is analytic in a simply-connected region of the z-plane containing the origin with the order
f (z) = O(|z|ε )
(z −→ 0),
with ε > max{0, γ − δ} − 1.
Here F (a, b; c; z) is the Gauss hypergeometric function defined by
F (a, b; c; z) =
∞
X
(a)n (b)n
n=0
(c)n (1)n
,
(27)
where (ν)n is the Pochhammer symbol defined by
Γ(ν + k)
=
(ν)n =
Γ(ν)
1
ν(ν + 1)(ν + 2) · · · (ν + n − 1)
(n = 0)
(n ∈ N)
(28)
and the multiplicity of (z − t)η−1 is removed by requiring log (z − t) to be real when
z − t > 0.
Remark 2. For γ = −η, we note that
η,−η,δ
I0,z
f (z) = Dz−η f (z).
In order to prove our result for the fractional integral operator, we have to recall
here the following lemma due to Srivastava et al. [14].
Lemma 2. If η > 0 and n > γ − δ − 1, then
η,γ,δ n
I0,z
z =
Γ(n + 1)Γ(n − γ + δ + 1) n−γ
z
.
Γ(n − γ + 1)Γ(n + η + δ + 1)
(29)
With aid of Lemma 2., we prove
Theorem 3. Let η > 0, γ > 2, η + δ > −2, γ − δ < 2, γ(η + δ) ≤ η(j + 2), and
j ∈ N. If f (z) ∈ BT (j, λ, α), then
128
B.A. Frasin - Some applications of fractional calculus operators...
(1 − α)(2 − γ + δ)j (2)j
Γ(2 − γ + δ) |z|1−γ
η,γ,δ
j
1−
|z|
I0,z f (z) ≥
Γ(2 − γ)Γ(2 + η + δ)
σ(j + 1, α, λ)(2 − γ)j (2 − γ + δ)j
(30)
and
(1 − α)(2 − γ + δ)j (2)j
Γ(2 − γ + δ) |z|1−γ
η,γ,δ
j
|z|
1+
I0,z f (z) ≤
Γ(2 − γ)Γ(2 + η + δ)
σ(j + 1, α, λ)(2 − γ)j (2 − γ + δ)j
(31)
for z ∈ U0 , where
U
(γ ≤ 1),
U0 =
(32)
U−{0}
(γ > 1).
The equalities in (30) and (31) are attained for the function f (z) given by (18).
Proof . By using Lemma 2, we have
Γ(2 − γ + δ)
z 1−γ
Γ(2 − γ)Γ(2 + η + δ)
∞
X
Γ(n + 1)Γ(n − γ + δ + 1)
an z n−γ
= −
Γ(n − γ + 1)Γ(n + η + δ + 1)
η,γ,δ
I0,z
f (z) =
(z ∈ U0 ).
n=j+1
Letting
Γ(2 − γ)Γ(2 + η + δ) γ η,γ,δ
z I0,z f (z)
Γ(2 − γ + δ)
∞
X
= z−
∆(n)an z n ,
Ω(z) =
(33)
n=j+1
where
∆(n) =
(2 − γ + δ)n−1 (2)n−1
(2 − γ)n−1 (2 + γ + δ)n−1
(n ≥ j + 1),
(34)
we can see that the function ∆(n) is non-increasing for integers n ≥ j + 1, then we
have
0 < ∆(n) ≤ ∆(j + 1) =
129
(2 − γ + δ)j (2)j
.
(2 − γ)j (2 + γ + δ)j
(35)
B.A. Frasin - Some applications of fractional calculus operators...
Therefore, by using (14) and (35), we have
|Ω(z)| ≥ |z| − ∆(j + 1) |z|j+1
∞
X
an
n=j+1
(1 − α)(2 − γ + δ)j (2)j
|z|j+1
≥ |z| −
σ(j + 1, α, λ)(2 − γ)j (2 + γ + δ)j
and
j+1
|Ω(z)| ≤ |z| + ∆(j + 1) |z|
∞
X
an
n=j+1
≤ |z| +
(1 − α)(2 − γ + δ)j (2)j
|z|j+1
σ(j + 1, α, λ)(2 − γ)j (2 + γ + δ)j
for z ∈ U0 , where U0 is defined by (32). This completes the proof of Theorem 3.
Remark 3. Taking γ = −η in Theorem 3, we get the result of Theorem 1.
4.Integral operators
Theorem 4. Let the functions f (z) defined by (3) be in the class BT (j, λ, α),
and c be a real number such that c > −1. Then the function F (z) defined by
Z
c + 1 z c−1
t f (t)dt
(c > −1)
(36)
F (z) =
zc
0
also belonging to the class BT (j, λ, α).
Proof. From (36) we have
∞
X
c+1
F (z) = z −
an z n .
c+n
n=j+1
Therefore,
∞
X
n=j+1
σ(n, α, λ)
c+1
c+n
an ≤
∞
X
σ(n, α, λ)an ≤ 1 − α,
n=j+1
since f (z) ∈ BT (j, λ, α). Hence, by Lemma 1, F (z) ∈ BT (j, λ, α).
Theorem 5. Let the function
130
B.A. Frasin - Some applications of fractional calculus operators...
F (z) = z −
∞
X
an z n
(an ≥ 0)
n=j+1
be in the class BT (j, λ, α) and let c be a real number such that c > −1. Then the
function given by (36) is univalent in |z| < R∗ , where
σ(n, α, λ)(c + 1)
R = R (n, α, c) = inf
n
n(1 − α)(c + n)
∗
∗
1/(n−1)
(n ≥ 2).
(37)
The result is sharp, with the function f (z) given by
f (z) = z −
(1 − α)(c + n) n
z
σ(n, α, λ)(c + 1)
(n ≥ 2).
(38)
Proof. From (36), we have
∞
X
z 1−c (z c F (z))′
c+n
f (z) =
=z−
an z n .
c+1
c+1
n=j+1
In order to obtain the required result, it suffices to show that
|f ′ (z) − 1| < 1 whenever |z| < R∗ , where R∗ is given by (37). Now
∞
X
′
n(c + n)
f (z) − 1 ≤
an |z|n−1 .
c+1
n=j+1
Thus |f ′ (z) − 1| < 1 if
∞
X
n(c + n)
an |z|n−1 < 1.
c+1
(39)
n=j+1
But from Lemma 1, (39) will be satisfied if
n(c + n)
σ(n, α, λ)
an |z|n−1 <
,
c+1
1−α
(40)
that is, if
|z| ≤
σ(n, α, λ)(c + 1)
n(1 − α)(c + n)
1/(n−1)
Therefore, f (z) is univalent in |z| < R∗ .
131
(n ≥ 2).
(41)
B.A. Frasin - Some applications of fractional calculus operators...
References
[1] M.K. Aouf, On fractional derivatives and fractional integrals of certain subclasses of starlike and convex functions, Math. Japon. 35 (1990), 831-837.
[2] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricemi, Tables of integral
Transforms, voll. II, McGraw-Hill Book Co., NewYork, Toronto and London,1954.
[3] B.A. Frasin, On the analytic functions with negative coefficients, Soochow J.
Math.Vol 31, No.3 (2005), 349-359.
[4] V.P. Gupta and P.K. Jain, Certain classes univalent analytic functions with
negative coefficients II, Bull. Austral. Math. Soc. 15 (1976), 467-473.
[5] K.B. Oldham and T. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integral to Arbitrary Order, Academic Press, NewYork
and London, 1974.
[6] S. Owa, On the distortion theorems I, Kyungpook Math. J. 18 (1978), 53-59.
[7] S. Owa, M. Saigo and H.M.Srivastava, Some characterization theorems for
starlike and convex functions involving a certain fractional integral operators, J.
Math. Anal. 140 (1989), 419-426.
[8] M. Saigo, A remark on integral operators involving the Gauss hypergeometric
functions, Math. Rep. College General Ed. Kyushu Univ. 11 (1978),135-143.
[9] S.G. Samko, A.A. Kilbas and O.I. Marchev, Integrals and Derivatives of Fractional Order and Some of Their Applications (Russian), Nauka i Teknika, Minsk,
1987.
[10] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc. 51 (1975),109-116.
[11] H.M.Srivastava and R.G. Buchman, Convolution Integral Equations with
Special Functions Kernels, John Wiely and Sons, NewYork, London, Sydney and
Toronto, 1977.
[12] H.M.Srivastava and S. Owa (Eds.), Univalent functions, Fractional Calculus,
and Their Applications,Halsted Press (Ellis Horworod Limited, Chichester ), John
Wiely and Sons, NewYork, Chichester, Brisbane and Toronto, 1989.
[13] H.M.Srivastava, S. Owa and S.K. Chatterjea, A note on certain classes of
starlike functions, Rend. Sem. Mat. Univ. Padova 77 (1987), 115-124.
[14] 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.
B.A. Frasin
Faculty of Science, Department of Mathematics
Al al-Bayt University, P.O. Box: 130095 Mafraq, Jordan
email:[email protected]
132
ISSN: 1582-5329
No. 23/2010
pp. 123-132
SOME APPLICATIONS OF FRACTIONAL CALCULUS
OPERATORS TO CERTAIN SUBCLASS OF ANALYTIC
FUNCTIONS WITH NEGATIVE COEFFICIENTS
B.A. Frasin
Abstract. The object of the present paper is to derive various distortion theorems for fractional calculus and fractional integral operators of functions in the class
BT(j, λ, α) consisting of analytic and univalent functions with negative coefficients.
Furthermore, some of integral operators of functions in the class BT(j, λ, α) is shown.
2000 Mathematics Subject Classification: 30C45.
1.Introduction and definitions
Let A(j) denote the family of functions of the form:
f (z) = z +
∞
X
an z n
(j ∈ N = {1, 2, 3, · · · }),
(1)
n=j+1
which are analytic in the open unit disk U = {z : |z| < 1}. A function f (z)
belonging to A(j) is in the class B(j, λ, α) if and only if
zf ′ (z) + (2λ2 − λ)z 2 f ′′ (z)
>α
(2)
Re
4(λ − λ2 )z + (2λ2 − λ)zf ′ (z) + (2λ2 − 3λ + 1)f (z)
for some α(0 ≤ α < 1) and λ(0 ≤ λ < 1), and for all z ∈ U.
Let T(j) denote the subclass of A(j) consisting of functions of the form:
f (z) = z −
∞
X
an z n
(an ≥ 0, j ∈ N),
(3)
n=j+1
Further, we define the class BT(j, λ, α) by
BT(j, λ, α) = B(j, λ, α) ∩ T(j).
123
(4)
B.A. Frasin - Some applications of fractional calculus operators...
The class BT(j, λ, α) was introduced and studied by the author in [3]. The
class BT(j, λ, α) is of special interest because it reduces to various classes of wellknown functions as well as many new ones. For example The classes BT(1, 0, α) =
T ∗ (α) and BT(1, 1, α) = C(α) were first studied by Silverman [10]. The classes
BT(j, 0, α) = Tα∗ (j) and BT(j, 1, α) = Cα (j) were studied Srivastava et al. [13]. The
class BT(1, 1/2, α) = BT(α) was studied by Gupta and Jain [4].
In order to show our results, we shall need the following lemma.
Lemma 1. ([3]) Let the function f (z) be defined by (3). Then f (z) ∈ BT(j, λ, α)
if and only if
∞
X
σ(n, α, λ)an ≤ 1 − α,
(5)
n=j+1
where
σ(n, α, λ) := (2λ2 − λ)n2 + [1 + (1 + α)(λ − 2λ2 )]n + (1 + 2λ2 − 3λ)α
(6)
and 0 ≤ α < 1, 0 ≤ λ < 1. The result is sharp.
2.Fractional calculus
Many essentially equivalent definitions of fractional calculus (that is fractional
derivatives and fractional integrals) have been given in the literature (cf., e.g., [1],
[2, Chap. 13], [5],[7], [8], [9], [11, p.28 et. seq.]. We find it to be convenient to recall
here the following definitions which are used earlier by Owa [6] (and, subsequently,
by Srivastava and Owa [12]).
Definition 1. The fractional integral of order µ is defined, for a function f (z),
by
Dz−µ f (z)
1
=
Γ(µ)
Zz
f (ζ)
dζ,
(z − ζ)1−µ
(7)
0
where µ > 0, f (z) is an analytic function in a simply-connected region of the z-plane
containing the origin, and the multiplicity of (z − ζ)1−µ is removed by requiring log
(z − ζ) to be real when z − ζ > 0.
Definition 2. The fractional derivative of order µ is defined, for a function
f (z), by
Dzµ f (z)
d
1
=
Γ(1 − µ) dz
Zz
0
124
f (ζ)
dζ,
(z − ζ)µ
(8)
B.A. Frasin - Some applications of fractional calculus operators...
where 0 ≤ µ < 1, f (z) is an analytic function in a simply-connected region of the
z-plane containing the origin, and the multiplicity of (z − ζ)−µ is removed as in
Definition 1 above.
Definition 3. Under the hypotheses of Definition 2, the fractional derivative of
order n + µ is defined by
dn µ
D f (z),
dz n z
where 0 ≤ µ < 1 and n ∈ N0 = {0, 1, 2, . . .}.
We begin by proving
Dzn+µ f (z) =
(9)
Theorem 1. If f (z) ∈ BT (j, λ, α), then
and
1+µ
−µ
Dz f (z) ≥ |z|
Γ(2 + µ)
1+µ
−µ
Dz f (z) ≤ |z|
Γ(2 + µ)
(10)
(11)
(1 − α)Γ(j + 2)Γ(2 + µ)
|z|j
1−
σ(j + 1, α, λ)Γ(j + 2 + µ)
(1 − α)Γ(j + 2)Γ(2 + µ)
|z|j
σ(j + 1, α, λ)Γ(j + 2 + µ)
1+
,
for µ > 0 and z ∈ U. The results (10) and (11) are sharp.
Proof . Define the function G(z)by
G(z) = Γ(2 + µ)z −µ Dz−µ f (z)
∞
X
Γ(n + 1)Γ(2 + µ)
= z−
an z n
Γ(n + 1 + µ)
= z−
n=j+1
∞
X
Ψ(n)an z n ,
n=j+1
where
Ψ(n) =
Γ(n + 1)Γ(2 + µ)
Γ(n + 1 + µ)
(n ≥ j + 1).
(12)
It easy to see that
0 < Ψ(n) ≤ Ψ(j + 1) =
125
Γ(j + 2)Γ(2 + µ)
.
Γ(j + 2 + µ)
(13)
B.A. Frasin - Some applications of fractional calculus operators...
Furthermore, it follows from Lemma 1 that
∞
X
an ≤
n=j+1
1−α
,
σ(j + 1, α, λ)
(14)
Therefore, by using (13) and (14), we can see that
|G(z)| ≥ |z| − Ψ(j + 1) |z|j+1
∞
X
an ≥ |z| −
n=j+1
(1 − α)Γ(j + 2)Γ(2 + µ)
|z|j+1 (15)
σ(j + 1, α, λ)Γ(j + 2 + µ)
and
|G(z)| ≤ |z|+Ψ(j +1) |z|j+1
∞
X
an ≤ |z|+
n=j+1
(1 − α)Γ(j + 2)Γ(2 + µ)
|z|j+1 , (16)
σ(j + 1, α, λ)Γ(j + 2 + µ)
which prove the inequalities of Theorem 1.
Finally, we can easily see that the results (10) and (11) are sharp for the function
f (z) given by
(1 − α)Γ(j + 2)Γ(2 + µ) j
z 1+µ
−µ
(17)
1−
z
Dz f (z) =
Γ(2 + µ)
σ(j + 1, α, λ)Γ(j + 2 + µ)
or
f (z) = z −
1−α
z j+1 .
σ(j + 1, α, λ)
(18)
Theorem 2. If f (z) ∈ BT (j, λ, α), then
|Dzµ f (z)|
|z|1−µ
≥
Γ(2 − µ)
|Dzµ f (z)|
|z|1−µ
≤
Γ(2 − µ)
and
(19)
(20)
(1 − α)Γ(j + 2)Γ(2 − µ)
|z|j
1−
σ(j + 1, α, λ)Γ(j + 2 − µ)
(1 − α)Γ(j + 2)Γ(2 − µ)
|z|j
1+
σ(j + 1, α, λ)Γ(j + 2 − µ)
for 0 ≤ µ < 1 and z ∈ U. The results (19) and (20) are sharp.
Proof. Define the function H(z) by
126
,
B.A. Frasin - Some applications of fractional calculus operators...
H(z) = Γ(2 − µ)z µ Dzµ f (z)
∞
X
Γ(n + 1)Γ(2 − µ)
= z−
an z n
Γ(n + 1 − µ)
= z−
n=j+1
∞
X
Φ(n)an z n ,
n=j+1
where
Γ(n)Γ(2 − µ)
Γ(n + 1 − µ)
Φ(n) =
(n ≥ j + 1).
(21)
Γ(j + 1)Γ(2 − µ)
.
Γ(j + 2 − µ)
(22)
It easy to see that
0 < Φ(n) ≤ Φ(j + 1) =
Consequently, with the aid of (14) and (22), we have
|H(z)| ≥ |z|−Φ(j +1) |z|j+1
∞
X
nan ≥ |z|−
n=j+1
(1 − α)Γ(j + 2)Γ(2 − µ)
|z|j+1 (23)
σ(j + 1, α, λ)Γ(j + 2 − µ)
and
j+1
|H(z)| ≤ |z| + Φ(j + 1) |z|
∞
X
nan ≤ |z| +
n=j+1
(1 − α)Γ(j + 2)Γ(2 − µ)
|z|j+1 .
σ(j + 1, α, λ)Γ(j + 2 − µ)
(24)
Now (19) and (20) follow from (23) and (24), respectively.
Finally, by taking the function f (z) defined by
(1 − α)Γ(j + 2)Γ(2 − µ) j
z 1−µ
1−
z
Dzµ f (z) =
Γ(2 − µ)
σ(j + 1, α, λ)Γ(j + 2 − µ)
(25)
or for the function given by (18), the results (19) and (20) are easily seen to be
sharp.
Remark 1. Letting µ = 0 in Theorem 1 and µ −→ 1 in Theorem 2, we shall
obtain the corresponding results Theorem 3 and Theorem 4 in [3].
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B.A. Frasin - Some applications of fractional calculus operators...
3.Fractional integral operator
We need the following definition of fractional integral operator given by Srivastava et al. [14].
Definition 4. For real number η > 0, γ and δ, the fractional integral operator
η,γ,δ
I0,z
is defined by
η,γ,δ
I0,z
f (z)
z −η−γ
=
Γ(η)
Zz
(z − t)η−1 F (η + γ, −δ; η; 1 − t/z)f (t)dt,
(26)
0
where a function f (z) is analytic in a simply-connected region of the z-plane containing the origin with the order
f (z) = O(|z|ε )
(z −→ 0),
with ε > max{0, γ − δ} − 1.
Here F (a, b; c; z) is the Gauss hypergeometric function defined by
F (a, b; c; z) =
∞
X
(a)n (b)n
n=0
(c)n (1)n
,
(27)
where (ν)n is the Pochhammer symbol defined by
Γ(ν + k)
=
(ν)n =
Γ(ν)
1
ν(ν + 1)(ν + 2) · · · (ν + n − 1)
(n = 0)
(n ∈ N)
(28)
and the multiplicity of (z − t)η−1 is removed by requiring log (z − t) to be real when
z − t > 0.
Remark 2. For γ = −η, we note that
η,−η,δ
I0,z
f (z) = Dz−η f (z).
In order to prove our result for the fractional integral operator, we have to recall
here the following lemma due to Srivastava et al. [14].
Lemma 2. If η > 0 and n > γ − δ − 1, then
η,γ,δ n
I0,z
z =
Γ(n + 1)Γ(n − γ + δ + 1) n−γ
z
.
Γ(n − γ + 1)Γ(n + η + δ + 1)
(29)
With aid of Lemma 2., we prove
Theorem 3. Let η > 0, γ > 2, η + δ > −2, γ − δ < 2, γ(η + δ) ≤ η(j + 2), and
j ∈ N. If f (z) ∈ BT (j, λ, α), then
128
B.A. Frasin - Some applications of fractional calculus operators...
(1 − α)(2 − γ + δ)j (2)j
Γ(2 − γ + δ) |z|1−γ
η,γ,δ
j
1−
|z|
I0,z f (z) ≥
Γ(2 − γ)Γ(2 + η + δ)
σ(j + 1, α, λ)(2 − γ)j (2 − γ + δ)j
(30)
and
(1 − α)(2 − γ + δ)j (2)j
Γ(2 − γ + δ) |z|1−γ
η,γ,δ
j
|z|
1+
I0,z f (z) ≤
Γ(2 − γ)Γ(2 + η + δ)
σ(j + 1, α, λ)(2 − γ)j (2 − γ + δ)j
(31)
for z ∈ U0 , where
U
(γ ≤ 1),
U0 =
(32)
U−{0}
(γ > 1).
The equalities in (30) and (31) are attained for the function f (z) given by (18).
Proof . By using Lemma 2, we have
Γ(2 − γ + δ)
z 1−γ
Γ(2 − γ)Γ(2 + η + δ)
∞
X
Γ(n + 1)Γ(n − γ + δ + 1)
an z n−γ
= −
Γ(n − γ + 1)Γ(n + η + δ + 1)
η,γ,δ
I0,z
f (z) =
(z ∈ U0 ).
n=j+1
Letting
Γ(2 − γ)Γ(2 + η + δ) γ η,γ,δ
z I0,z f (z)
Γ(2 − γ + δ)
∞
X
= z−
∆(n)an z n ,
Ω(z) =
(33)
n=j+1
where
∆(n) =
(2 − γ + δ)n−1 (2)n−1
(2 − γ)n−1 (2 + γ + δ)n−1
(n ≥ j + 1),
(34)
we can see that the function ∆(n) is non-increasing for integers n ≥ j + 1, then we
have
0 < ∆(n) ≤ ∆(j + 1) =
129
(2 − γ + δ)j (2)j
.
(2 − γ)j (2 + γ + δ)j
(35)
B.A. Frasin - Some applications of fractional calculus operators...
Therefore, by using (14) and (35), we have
|Ω(z)| ≥ |z| − ∆(j + 1) |z|j+1
∞
X
an
n=j+1
(1 − α)(2 − γ + δ)j (2)j
|z|j+1
≥ |z| −
σ(j + 1, α, λ)(2 − γ)j (2 + γ + δ)j
and
j+1
|Ω(z)| ≤ |z| + ∆(j + 1) |z|
∞
X
an
n=j+1
≤ |z| +
(1 − α)(2 − γ + δ)j (2)j
|z|j+1
σ(j + 1, α, λ)(2 − γ)j (2 + γ + δ)j
for z ∈ U0 , where U0 is defined by (32). This completes the proof of Theorem 3.
Remark 3. Taking γ = −η in Theorem 3, we get the result of Theorem 1.
4.Integral operators
Theorem 4. Let the functions f (z) defined by (3) be in the class BT (j, λ, α),
and c be a real number such that c > −1. Then the function F (z) defined by
Z
c + 1 z c−1
t f (t)dt
(c > −1)
(36)
F (z) =
zc
0
also belonging to the class BT (j, λ, α).
Proof. From (36) we have
∞
X
c+1
F (z) = z −
an z n .
c+n
n=j+1
Therefore,
∞
X
n=j+1
σ(n, α, λ)
c+1
c+n
an ≤
∞
X
σ(n, α, λ)an ≤ 1 − α,
n=j+1
since f (z) ∈ BT (j, λ, α). Hence, by Lemma 1, F (z) ∈ BT (j, λ, α).
Theorem 5. Let the function
130
B.A. Frasin - Some applications of fractional calculus operators...
F (z) = z −
∞
X
an z n
(an ≥ 0)
n=j+1
be in the class BT (j, λ, α) and let c be a real number such that c > −1. Then the
function given by (36) is univalent in |z| < R∗ , where
σ(n, α, λ)(c + 1)
R = R (n, α, c) = inf
n
n(1 − α)(c + n)
∗
∗
1/(n−1)
(n ≥ 2).
(37)
The result is sharp, with the function f (z) given by
f (z) = z −
(1 − α)(c + n) n
z
σ(n, α, λ)(c + 1)
(n ≥ 2).
(38)
Proof. From (36), we have
∞
X
z 1−c (z c F (z))′
c+n
f (z) =
=z−
an z n .
c+1
c+1
n=j+1
In order to obtain the required result, it suffices to show that
|f ′ (z) − 1| < 1 whenever |z| < R∗ , where R∗ is given by (37). Now
∞
X
′
n(c + n)
f (z) − 1 ≤
an |z|n−1 .
c+1
n=j+1
Thus |f ′ (z) − 1| < 1 if
∞
X
n(c + n)
an |z|n−1 < 1.
c+1
(39)
n=j+1
But from Lemma 1, (39) will be satisfied if
n(c + n)
σ(n, α, λ)
an |z|n−1 <
,
c+1
1−α
(40)
that is, if
|z| ≤
σ(n, α, λ)(c + 1)
n(1 − α)(c + n)
1/(n−1)
Therefore, f (z) is univalent in |z| < R∗ .
131
(n ≥ 2).
(41)
B.A. Frasin - Some applications of fractional calculus operators...
References
[1] M.K. Aouf, On fractional derivatives and fractional integrals of certain subclasses of starlike and convex functions, Math. Japon. 35 (1990), 831-837.
[2] A. Erdélyi, W. Magnus, F. Oberhettinger and F.G. Tricemi, Tables of integral
Transforms, voll. II, McGraw-Hill Book Co., NewYork, Toronto and London,1954.
[3] B.A. Frasin, On the analytic functions with negative coefficients, Soochow J.
Math.Vol 31, No.3 (2005), 349-359.
[4] V.P. Gupta and P.K. Jain, Certain classes univalent analytic functions with
negative coefficients II, Bull. Austral. Math. Soc. 15 (1976), 467-473.
[5] K.B. Oldham and T. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integral to Arbitrary Order, Academic Press, NewYork
and London, 1974.
[6] S. Owa, On the distortion theorems I, Kyungpook Math. J. 18 (1978), 53-59.
[7] S. Owa, M. Saigo and H.M.Srivastava, Some characterization theorems for
starlike and convex functions involving a certain fractional integral operators, J.
Math. Anal. 140 (1989), 419-426.
[8] M. Saigo, A remark on integral operators involving the Gauss hypergeometric
functions, Math. Rep. College General Ed. Kyushu Univ. 11 (1978),135-143.
[9] S.G. Samko, A.A. Kilbas and O.I. Marchev, Integrals and Derivatives of Fractional Order and Some of Their Applications (Russian), Nauka i Teknika, Minsk,
1987.
[10] H. Silverman, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc. 51 (1975),109-116.
[11] H.M.Srivastava and R.G. Buchman, Convolution Integral Equations with
Special Functions Kernels, John Wiely and Sons, NewYork, London, Sydney and
Toronto, 1977.
[12] H.M.Srivastava and S. Owa (Eds.), Univalent functions, Fractional Calculus,
and Their Applications,Halsted Press (Ellis Horworod Limited, Chichester ), John
Wiely and Sons, NewYork, Chichester, Brisbane and Toronto, 1989.
[13] H.M.Srivastava, S. Owa and S.K. Chatterjea, A note on certain classes of
starlike functions, Rend. Sem. Mat. Univ. Padova 77 (1987), 115-124.
[14] 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.
B.A. Frasin
Faculty of Science, Department of Mathematics
Al al-Bayt University, P.O. Box: 130095 Mafraq, Jordan
email:[email protected]
132