Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 2, Article 28, 2004
INEQUALITIES DEFINING CERTAIN SUBCLASSES OF ANALYTIC AND
MULTIVALENT FUNCTIONS INVOLVING FRACTIONAL CALCULUS
OPERATORS
R.K. RAINA AND I.B. BAPNA
D EPARTMENT O F M ATHEMATICS
M.P. U NIVERSITY O F AGRI . & T ECHNOLOGY
C OLLEGE O F T ECHNOLOGY A ND E NGINEERING
U DAIPUR 313001, R AJASTHAN , I NDIA .
rainark_7@hotmail.com
D EPARTMENT O F M ATHEMATICS ,
G OVT. P OSTGRADUATE C OLLEGE
B HILWARA 311001
R AJASTHAN , I NDIA .
bapnain@yahoo.com
Received 25 July, 2003; accepted 09 February, 2004
Communicated by N.E. Cho
A BSTRACT. Making use of a certain fractional calculus operator, we introduce two new subclasses Mδ (p; λ, µ, η) and Tδ (p; λ, µ, η) of analytic and p−valent functions in the open unit
disk. The results investigated exhibit the sufficiency conditions for a function to belong to each
of these classes. Several geometric properties of such multivalent functions follow, and these
consequences are also mentioned.
Key words and phrases: Analytic functions, Multivalent functions, Starlike functions, Convex functions, Fractional calculus
operators.
2000 Mathematics Subject Classification. 30C45, 26A33.
1. I NTRODUCTION AND D EFINITIONS
Let Ap denote the class of functions of the form
(1.1)
f (z) = z p +
∞
X
an+p z n+p
(p ∈ N = {1, 2, 3, . . . }) ,
n=1
which are analytic and p−valent in the open unit disk U = {z : z ∈ C and |z| < 1}.
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
This work was supported by Council for Scientific and Industrial Research, India.
The authors express their sincerest thanks to the referee for suggestions.
102-03
2
R.K. R AINA AND I.B. BAPNA
A function f (z) ∈ Ap is said to be p−valently starlike in U, if
0 zf (z)
(1.2)
<
>0
(z ∈ U),
f (z)
and the function f (z) ∈ Ap is said to be p−valently convex in U, if
zf 00 (z)
(1.3)
< 1+ 0
>0
(z ∈ U).
f (z)
Further, a function f (z) ∈ Ap is said to be p−valently close-to-convex in U, if
0 f (z)
(1.4)
<
>0
(z ∈ U).
z p−1
One may refer to [1], [2] and [9] for above definitions and other related details.
λ,µ,η
The operator J0,z
occurring in this paper is the Saigo type fractional calculus operator
defined as follows ([8]):
Definition 1.1. Let 0 ≤ λ < 1 and µ,η ∈ R, then
λ−µ Z z
z
t
d
λ,µ,η
−λ
(1.5) J0,z f (z) =
(z − t) F1 µ − λ, 1 − η; 1 − λ; 1 −
f (t)dt ,
dz Γ(1 − λ) 0
z
where the 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), where ε > max{0, µ − η} − 1.
It being understood that (z − t)−λ denotes the principal value for 0 ≤ arg(z − t) < 2π.
The 2 F1 function occurring in the right-hand side of (1.5) is the familiar Gaussian hypergeometric function (see [9] for its definition).
Definition 1.2. Under the hypotheses of Definition 1.1, a fractional calculus operator
λ+m,µ+m,η+m
J0,z
is defined by ([7])
(1.6)
λ+m,µ+m,η+m
J0,z
f (z) =
dm λ,µ,η
J
f (z) (z ∈ U; m ∈ N0 = {0} ∪ N) .
dz m 0,z
We observe that
(1.7)
λ,λ,η
Dzλ f (z) = J0,z
f (z)
(0 ≤ λ < 1),
and
(1.8)
λ+m,λ+m,η+m
Dzλ+m f (z) = J0,z
f (z)
(0 ≤ λ < 1; m ∈ N0 ),
where Dzλ+m is the well known fractional derivative operator ([6], [9]).
We introduce here two subclasses of functions Mδ (p; λ, µ, η) and Tδ (p; λ, µ, η) which are
defined as follows.
Definition 1.3. Let δ ∈ R \ {0}, p ∈ N, 0 ≤ λ < 1, µ < 1, and η > max(λ, µ) − p − 1. Then
the function f (z) ∈ Ap is said to belong to Mδ (p; λ, µ, η) if it satisfies the inequality
!δ
λ+1,µ+1,η+1
zJ0,z
f (z)
δ
− (p − µ) < (p − µ)δ
(1.9)
(z ∈ U),
λ,µ,η
J0,z f (z)
δ
λ+1,µ+1,η+1
λ,µ,η
f (z)/J0,z f (z) is taken as its principal value.
where the value of z J0,z
J. Inequal. Pure and Appl. Math., 5(2) Art. 28, 2004
http://jipam.vu.edu.au/
I NEQUALITIES D EFINING S UBCLASSES OF A NALYTIC AND M ULTIVALENT F UNCTIONS
3
Definition 1.4. Under the hypotheses of Definition 1.3, the function f (z) ∈ Ap is said to belong
to Tδ (p; λ, µ, η) if it satisfies the inequality
δ Γ(p + 1)Γ(p + η − µ + 1) δ µ−p λ,µ,η
(1.10) z J0,z f (z) −
Γ(p − µ + 1)Γ(p + η − λ + 1) δ
Γ(p + 1)Γ(p + η − µ + 1)
<
(z ∈ U),
Γ(p − µ + 1)Γ(p + η − λ + 1)
δ
λ,µ,η
where the value of z µ−p J0,z
f (z) is considered to be its principal value. For λ = µ, we
have
(1.11)
Mδ (p; µ, µ, η) = Mδ (p; µ),
and
(1.12)
Tδ (p; µ, µ, η) = Tδ (p; µ).
The classes Mδ (p; µ)and Tδ (p; µ) were studied recently in [4]. In view of the operational
relation (1.8), it may be noted that the functions in M1 (p; 0) are p−valently starlike in U,
whereas, the functions inT1 (p; 1) are p−valently close-to-convex in U.
In this paper we investigate characterization properties giving sufficiency conditions for functions of the form (1.1) to belong to the classes Mδ (p; λ, µ, η) and Tδ (p; λ, µ, η) involving the
fractional calculus operator (1.6). Several consequences of the main results and their relevance
to known results are also pointed out.
2. R ESULTS R EQUIRED
We mention the following results which are used in the sequel:
Lemma 2.1. ([8]). If 0 ≤ λ < 1; µ,η ∈ R and k > max{0, µ − η} − 1, then
Γ(1 + k)Γ(1 − µ + η + k) k−µ
λ,µ,η k
z .
(2.1)
J0,z
z =
Γ(1 − µ + k)Γ(1 − λ + η + k)
Lemma 2.2. ([5]). Let w(z) be an analytic function in the unit disk U with w(0) = 0, and let
0 < r < 1. If |w(z)| attains at z0 its maximum value on the circle |z| = r, then
(2.2)
z0 w0 (z0 ) = kw(z0 )
(k ≥ 1).
3. M AIN R ESULTS
We begin by proving
Theorem 3.1. Let δ ∈ R \ {0}, p ∈ N, 0 ≤ λ < 1, µ < 1, η > max(λ, µ) − p − 1, and
a > 0, b ≥ 0, such that a + 2b ≤ 1. If a function f (z) ∈ Ap satisfies the inequality
!#
"
λ+1,µ+1,η+1
λ+2,µ+2,η+2
J0,z
f (z) J0,z
f (z)
−
(3.1) < 1 + z
λ+1,µ+1,η+1
λ,µ,η
J0,z
f (z)
J0,z
f (z)

a+b

<
(δ > 0)



δ(1 + a)(1 − b)
(z ∈ U),


a
+
b

 >
(δ < 0)
δ(1 + a)(1 − b)
then f (z) ∈ Mδ (p; λ, µ, η).
J. Inequal. Pure and Appl. Math., 5(2) Art. 28, 2004
http://jipam.vu.edu.au/
4
R.K. R AINA AND I.B. BAPNA
Proof. Let f (z) ∈ Ap , and define a function w(z) by
!δ
λ+1,µ+1,η+1
zJ0,z
f (z)
1 + aw(z)
δ
(3.2)
= (p − µ)
λ,µ,η
1 − bw(z)
J0,z
f (z)
(z ∈ U).
Then it follows from (2.1) that w(z) is analytic function in U, and w(0) = 0. Differentiation of
(3.2) gives
(
!)
λ+2,µ+2,η+2
λ+1,µ+1,η+1
J0,z
f (z) J0,z
f (z)
(a + b)zw0 (z)
1
(3.3) 1 + z
−
=
λ+1,µ+1,η+1
λ,µ,η
δ (1 + aw(z)) (1 − bw(z))
J0,z
f (z)
J0,z
f (z)
= φ(z) (say).
Assume that there exists a point z0 ∈ U such that
max |w(z)| = |w(z0 )| = 1.
|z|≤|z0 |
Then, applying Lemma 2.2, we can write
z0 w0 (z0 ) = kw(z0 )
(k ≥ 1),
and w(z0 ) = eiθ (θ ∈ [0, 2π)), so that from (3.3) we have
w(z0 )
k(a + b)
<{φ(z0 )} =
<
δ
(1 + aw(z0 )) (1 − bw(z0 ))
k
1
1
= <
−
δ
1 − bw(z0 ) 1 + aw(z0 )
k
1 − be−iθ
1 + ae−iθ
−
= <
δ
1 + b2 − 2b cos θ 1 + a2 + 2a cos θ
)
(
k∆
k
1
1
=
,
−
=
2 −1
2 −1
a
b
δ
δ 2 + 1−b cos θ
2 + 1+a cos θ
where θ 6= cos−1 (−1/a) and θ 6= cos−1 (−1/b).
Simple calculations (under the constraints mentioned with the hypotheses for the parameters
(a+b)
, and since k ≥ 1, it follows that
a and b) yield that ∆ ≥ (1+a)(1−b)

(a+b)

> δ(1+a)(1−b)
(δ > 0),

k∆
(3.4)
<{φ(z0 )} =
δ 
 < (a+b)
(δ < 0).
δ(1+a)(1−b)
This contradicts our condition (3.1), and we conclude from (3.2) that
!δ
λ+1,µ+1,η+1
(a + b)w(z) zJ0,z
f (z)
δ
δ
− (p − µ) = (p − µ) λ,µ,η
1 − bw(z) J0,z
a+b
δ
< (p − µ)
≤ (p − µ)δ .
1−b
This completes the proof of Theorem 3.1.
Next we prove
J. Inequal. Pure and Appl. Math., 5(2) Art. 28, 2004
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I NEQUALITIES D EFINING S UBCLASSES OF A NALYTIC AND M ULTIVALENT F UNCTIONS
5
Theorem 3.2. Let δ ∈ R \ {0}, p ∈ N, 0 ≤ λ < 1, µ < 1, η > max(λ, µ) − p − 1, and a > 0,
b ≥ 0 such that a + 2b ≤ 1. If a function f (z) ∈ Ap satisfies the inequality
a+b
! < p − µ +
(δ > 0)
λ+1,µ+1,η+1
δ(1+a)(1−b)
zJ0,z
f (z) 
(3.5)
<
(z ∈ U),
λ,µ,η

a+b
J0,z
> p − µ + δ(1+a)(1−b)
(δ > 0)
then f (z) ∈ Tδ (p; λ, µ, η).
Proof. Consider
δ Γ(1 + p)Γ(1 + p + η − µ) δ 1 + aw(z) µ−p λ,µ,η
(3.6)
z J0,z f (z) =
(z ∈ U).
Γ(1 + p − µ)Γ(1 + p + η − λ)
1 − bw(z)
Using the same method as elucidated in the proof of Theorem 3.1, we arrive at the desired
result.
Remark 3.3. If we set λ = µ, a = 1, b = 0, then Theorems 3.1 and 3.2 by appealing to the
operational relation (1.8) correspond to the recently established results due to Irmak et al. [4,
pp. 271–272].
Theorems 3.1 and 3.2 would also yield various results involving analytic and multivalent
functions by suitably choosing the values of a, b, δ, µ and p. Setting δ = 1 in Theorems 3.1 and
3.2, we have
Corollary 3.4. Let p ∈ N, 0 ≤ λ < 1, µ < 1, η > max(λ, µ) − p − 1, and a > 0, b ≥ 0 such
that a + 2b ≤ 1. If a function f (z) ∈ Ap satisfies the inequality
!)
(
λ+1,µ+1,η+1
λ+2,µ+2,η+2
f (z)
J0,z
f (z) J0,z
a+b
(3.7) < 1 + z
−
<
(z ∈ U),
λ+1,µ+1,η+1
λ,µ,η
(1 + a)(1 − b)
J0,z
f (z)
J0,z f (z)
then f (z) ∈ M1 (p; λ, µ, η).
Corollary 3.5. Let p ∈ N, 0 ≤ λ < 1, µ < 1, η > max(λ, µ) − p − 1, and a > 0, b ≥ 0 such
that a + 2b ≤ 1. If a function f (z) ∈ Ap satisfies the inequality
!
λ+1,µ+1,η+1
zJ0,z
f (z)
a+b
<p−µ+
(3.8)
<
(z ∈ U),
λ,µ,η
(1 + a)(1 − b)
J0,z f (z)
then f (z) ∈ T1 (p; λ, µ, η).
Corollaries 3.4 and 3.5 on putting λ = µ = 0, and using (1.8) give the following results:
Corollary 3.6. Let p ∈ N, a > 0, b ≥ 0 such that a + 2b ≤ 1. If a function f (z) ∈ Ap satisfies
the inequality
zf 00 (z) zf 0 (z)
a+b
(3.9)
< 1+ 0
−
<
(z ∈ U),
f (z)
f (z)
(1 + a)(1 − b)
then f (z) is p-valently starlike in U.
Corollary 3.7. Let p ∈ N, a > 0, b ≥ 0 such that a + 2b ≤ 1. If a function f (z) ∈ Ap satisfies
the inequality
zf 0(z)
a+b
(3.10)
<
<p+
(z ∈ U),
f (z)
(1 + a)(1 − b)
n
o
f (z)
then < zp > 0, (z ∈ U).
J. Inequal. Pure and Appl. Math., 5(2) Art. 28, 2004
http://jipam.vu.edu.au/
6
R.K. R AINA AND I.B. BAPNA
Lastly, Corollaries 3.4 and 3.5 on putting λ = µ = 1, and using (1.8) give
Corollary 3.8. Let p ∈ N, a > 0, b ≥ 0 such that a + 2b ≤ 1. If a function f (z) ∈ Ap satisfies
the inequality
zf 000 (z) zf 00 (z)
a+b
(3.11)
< 1 + 00
− 0
<
(z ∈ U),
f (z)
f (z)
(1 + a)(1 − b)
then f (z) is p−valently convex in U.
Corollary 3.9. Let p ∈ N, a > 0, b ≥ 0 such that a + 2b ≤ 1. If a function f (z) ∈ Ap satisfies
the inequality
00 zf (z)
a+b
(3.12)
<
<p−1+
(z ∈ U),
0
f (z)
(1 + a)(1 − b)
then f (z) is p−valently close-to -convex in U.
Remark 3.10. When a = 1, b = 0, then the Corollaries 3.6 – 3.9 correspond to the known
results [3, pp. 457–458] involving inequalities on p−valent functions.
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J. Inequal. Pure and Appl. Math., 5(2) Art. 28, 2004
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