Bulletin of Mathematical analysis and Applications
ISSN:1821-1291, URL: http://www.bmathaa.org
Volume 1, Issue 1, (2009) Pages 1-14
Some applications of differential subordination to
a general class of multivalently analytic functions
involving a convolution structure ∗
J. K. Prajapat & R. K. Raina
Abstract
By appealing to the familiar convolution structure of analytic functions,
we investigate various useful properties and characteristics of a general
class of multivalently analytic functions using the techniques of differential subordination. Several results are presented exhibiting relevant connections with some of the results presented here and those obtained in
earlier works.
1
Introduction
Let Ap denote the class of functions of the form
f (z) = z p +
∞
X
ak z k
(p ∈ N = {1, 2, ...}),
(1.1)
k=p+1
which are analytic and p−valent in the open unit disk U = {z; z ∈ C : |z| < 1}.
We denote by P(γ) the class of functions φ(z) of the form
φ(z) = 1 + c1 z + c2 z 2 + ...
(1.2)
which are analytic in U and satisfy the following inequality:
<(φ(z)) > γ
(0 ≤ γ < 1; z ∈ U).
Let the functions f and g be analytic in U, then we say that f is subordinate
to g in U, and write f ≺ g, if there exists a function w(z) analytic in U such
that |w(z)| < 1, z ∈ U, and w(0) = 0 with f (z) = g(w(z)) in U. In particular,
∗ Mathematics Subject Classifications: Primary 30C45; Secondary 26A33, 33C20.
Key words: Multivalently analytic functions, Hadamard product (or convolution),
Differential subordination, Hypergeometric functions, Fractional Differintegral operator,
Ruscheweyh derivative operator.
c
2009
Universiteti i Prishtines, Prishtine, Kosovë.
Submitted: November 2008, Published January 2009
1
2
Some applications of differential subordination
if f is univalent in U, then f ≺ g is equivalent to f (0) = g(0) and f (U) ⊂ g(U).
If f ∈ Ap is given by (1.1) and g ∈ Ap is given by
p
g(z) = z +
∞
X
bk z k ,
(1.3)
k=p+1
then the Hadamard product (or convolution) f ∗ g of f and g is defined (as
usual) by
∞
X
p
(f ∗ g)(z) := z +
ak bk z k .
(1.4)
k=p+1
For a given function g(z) ∈ Ap (defined by (1.3)), we define here a class
Jp (g; α, A, B) of functions belonging to the subclasses of Ap which consist of
functions f (z) of the form (1.1) and satisfying the following subordination:
(1 − α)
1 + Az
(f ∗ g)(z) α (f ∗ g)0 (z)
+
≺
zp
p
z p−1
1 + Bz
(1.5)
(z ∈ U; α > 0; −1 ≤ B < A ≤ 1).
Since
<
1 + Az
1 + Bz
>
1−A
1−B
(z ∈ U; −1 ≤ B < A ≤ 1),
(1.6)
therefore, by choosing
A=1−
2β
p
(0 ≤ β < p)
and
B = −1,
in subordination relation (1.5), we obtain the following inequality
(f ∗ g)(z) α (f ∗ g)0 (z)
β
+
(z ∈ U; α > 0; 0 ≤ β < p).
< (1 − α)
>
zp
p
z p−1
p
(1.7)
We further let Jp∗ (g; α, β) denote the class of functions f ∈ Ap which satisfies
the inequality (1.7).
Our aim in this paper is to investigate various useful and interesting properties of the function classes Jp (g; α, A, B) and Jp∗ (g; α, β) (defined above) involving the Hadamard product of two multivalently analytic functions and the
pricipal of subordination. Several corollaries are deduced from the main results
and their connections with known results are also pointed out. The concluding
section exhibits relevant connections to some of the results presented here and
those obtained in earlier works.
2
Preliminaries and Key Lemmas
We require the following lemmas to investigate the function class Jp (g; α, A, B)
(defined above).
J. K. Prajapat & R. K. Raina
3
Lemma 1 ([5, p. 71]). Let h(z) be a convex (univalent) function in U with
h(0) = 1, and let the function φ(z) of the form (1.2) be analytic in U. If
φ(z) +
then
zφ0 (z)
≺ h(z)
γ
γ
φ(z) ≺ ψ(z) := γ
z
Z
(<(γ) ≥ 0 (γ 6= 0); z ∈ U),
(2.1)
z
tγ−1 h(t) dt ≺ h(z)
(z ∈ U),
(2.2)
0
and ψ(z) is the best dominant.
Lemma 2([14, Lemma 1]). Let φ(z) of the form (1.2) be analytic in U. If
<(φ(z) + η zφ0 (z)) > β
(<(η) ≥ 0 (η 6= 0); β < 1; z ∈ U),
then
< (φ(z)) > β + (1 − β)(2 Υ − 1),
where Υ is given by
Z
1
Υ = Υ<(η) =
1 + t <(η)
−1
dt.
0
Lemma 3 ([13]). If φ(z) ∈ P(γ) (0 ≤ γ < 1), then
<(φ(z)) ≥ 2γ − 1 +
2(1 − γ)
1 + |z|
(z ∈ U).
Lemma 4 ([20], see also [17, Lemma 3]). For 0 ≤ γ1 , γ2 < 1,
P(γ1 ) ∗ P(γ2 ) ⊂ P(γ3 )
(γ3 = 1 − 2(1 − γ1 )(1 − γ2 )).
The result γ3 is the best possible.
Lemma 5 ([5, p. 35]). Suppose that the function ψ : C2 × U → C satisfies the
condition
<(ψ(ix, y; z)) ≤ ε
for some ε > 0 , real x, y ≤ −(1 + x2 )/2 for all z ∈ U. If φ(z) given by
(1.2) is analytic in U and
<(ψ(φ(z), zφ0 (z); z)) > ε,
then
<(φ(z)) > 0
(z ∈ U).
The generalized hypergeometric function p Fq is defined by(cf., e. g. [16, p.
333])
p Fq (z) ≡ p Fq (α1 , ..., αr ; β1 , ..., βs ; z)
4
Some applications of differential subordination
=:
∞
X
(α1 )n ... (αq )n z n
.
(β1 )n ... (βs )n n!
n=0
(2.3)
(z ∈ U; αj ∈ C(j = 1, ..., q), βj ∈ C\{0, −1, −2, ...}(j = 1, ..., s), q ≤ s+1; q, s ∈ N0 ),
where the symbol (α)k is the familiar Pochhammer symbol defined by
(α)k = α(α + 1)...(α + k − 1); k ∈ N.
(α)0 = 1,
Each of the following identities (asserted by Lemma 6) below are quite well
known (see [1, pp. 556-558]).
Lemma 6. For real or complex numbers a, b and c (c 6= 0, −1, −2, ...) :
1
Z
tb−1 (1−t)c−b−1 (1−zt)−a dt =
0
Γ(b) Γ(c − b)
2 F1 (a, b; c; z)
Γ(c)
(<(c) > <(b) > 0);
(2.4)
−a
(a, b; c; z) = (1 − z)
2 F1
2 F1
z
a, c − b; c;
z−1
;
(2.5)
(b + 1) 2 F1 (1, b; b + 1; z) = (b + 1) + b z 2 F1 (1, b + 1; b + 2; z)
3
(2.6)
Main Results
Our first main result is given by Theorem 1 below.
Theorem 1. Let f (z) ∈ Ap ,
Q0 f (z) =
Qµn f (z)
µ+1
= µ+1
z
Z
f (z)
zp
(z ∈ U),
(3.1)
z
tµ Qµn−1 f (t) dt
(µ > −1; n ∈ N0 ; z ∈ U),
(3.2)
0
where Qµ0 f (z) = Q0 f (z). If f (z) ∈ Jp (g; α, A, B), then (for |z| = r < 1)
< (Qµn (f ∗ g)(z)) ≥ ρn (r) > ρn (1)
(n ∈ N),
(3.3)
where
0 < ρn (r) = 1 + (B − A)(µ + 1)n p
∞
X
k=1
B k−1 rk
< 1.
(p + αk) (k + µ + 1)n
(3.4)
The estimate (3.3) is sharp.
Proof. We shall prove this theorem by the principal of mathematical induction
on n. Let f (z) ∈ Jp (g; α, A, B), and assume that
(f ∗ g)(z)
= q(z).
zp
(3.5)
J. K. Prajapat & R. K. Raina
5
It is clear that q(z) is of the form (1.2) and is analytic in U with q(0) = 1.
Differenting (3.5) with respect to z, and after some computation, we obtain
(1 − α)
(f ∗ g)(z) α (f ∗ g)0 (z)
+
zp
p
z p−1
= q(z) +
≺
Now, by using Lemma 1 for γ =
α
z q 0 (z)
p
1 + Az
1 + Bz
(z ∈ U).
p
α
, we deduce that
Z z
p
(f ∗ g)(z)
p −p
1 + At
α
t α −1
≺ Ω(z) = z
dt.
zp
α
1
+ Bt
0
(3.6)
Since −1 ≤ B < A ≤ 1 and α > 0, it follows from (3.6) that for
|z| = r < 1 :
Z 1
p
p
1 + Auz
−1
α
u
du
<(q(z)) ≥
<
α 0
1 + Buz
→ Ω(−r),
as z → −r.
(3.7)
Using Lemma 6 in (3.7), we get
p
p
(f ∗ g)(z)
2 F1 1, α ; α + 1; Br −
<
≥ ρ0 (r) =
p
Ar
1 − α+p
zp
p
p+α Ar 2 F1
1, αp + 1;
p
α
+ 2; Br
Simplifying right-hand side of the above estimate, we infer that
(
P∞
k−1 k
(f ∗ g)(z)
(B =
6 0);
1 + (B − A)p k=1 Bp+αkr
<
≥ ρ0 (r) =
p
zp
1 − α+p
Ar
(B = 0),
which implies that (3.3) holds true for n = 0.
For n = 1, we find that
Z
µ + 1 z µ (f ∗ g)(t)
< (Qµ1 (f ∗ g)(z)) = <
t
dt
z µ+1 0
tp
Z
µ+1 r µ
(f ∗ g)(ueiθ )
=
u
<
du
rµ+1 0
(ueiθ )p
!
Z
∞
X
µ+1 r µ
B k−1 uk
du
≥
u
1 + (B − A)p
rµ+1 0
p + αk
k=1
!
Z r X
∞
µ+1
B k−1 uµ+k
= 1 + µ+1 (B − A)p
du.
r
p + αk
0
k=1
We note that for |B| ≤ 1, u < 1 and p + αk ≥ p + α (k ≥ 1), the series in
the right-hand side is uniformly convergent in U, so that it can be integrated
term by term. Thus, we have
< (Qµ1 (f ∗ g)(z)) ≥ ρ1 (r) = 1 + (B − A)(µ + 1)p
∞
X
k=1
B k−1 rk
,
(p + αk) (k + µ + 1)
(B 6= 0);
(B = 0) .
6
Some applications of differential subordination
and this shows that (3.3) is also true for n = 1.
Next, we assume that (3.3) holds true for n = m. By letting t = ueiθ , we
obtain
Z
µ+1 z µ µ
µ
t (Qm (f ∗ g)(t)) dt
< Qm+1 (f ∗ g)(z) = <
z µ+1 0
Z r
µ+1
=
uµ < Qµm (f ∗ g)(ueiθ ) du
µ+1
r
0
!
Z
∞
X
µ+1 r µ
B k−1 uk
m
≥
u
1 + (B − A)(µ + 1) p
du
rµ+1 0
(p + αk) (k + µ + 1)m
k=1
!
Z r X
∞
k−1 µ+k
B
u
µ+1
du.
= 1 + µ+1 (B − A)(µ + 1)m+1 p
r
(p + αk) (k + µ + 1)m+1
0
k=1
Noting that the integrand is uniformly convergent in U, we deduce that
∞
X
< Qµm+1 (f ∗ g)(z) ≥ ρm+1 (r) = 1+(B−A)(µ+1)m+1 p
k=1
B k−1 rk
.
(p + αk) (k + µ + 1)m+1
Therefore, we conclude that
< (Qµn (f ∗ g)(z)) ≥ ρn (r),
for any integer n ∈ N0 .
Finally to prove the sharpness of the result (3.3), let us consider the function
Gn (r) = 1 + (B − A)(µ + 1)n p
∞
X
k=1
B k−1 rk
(p + αk) (k + µ + 1)n
(0 < r < 1).
The series Gn (r) is absolutely and uniformly convergent for each n ∈ N0 and
0 < r < 1. By suitably rearranging the terms of Gn (r), it is easy to see that
0 < Gn (r) < 1. Further since Gn (r) ≤ Gn−1 (r) and
Z r
rµ+1 Gn (r) = (µ + 1)
uµ Gn−1 (u) du
(n ∈ N),
0
we assert that Gn0 (r) ≤ 0, and that Gn (r) decreases with r as r → 1−
for fixed n, and increases to 1 as n → ∞ for fixed r. This implies that
Gn (r) > Gn (1). Therefore the estimate in (3.3) is sharp. This completes the
proof of Theorem 1.
Setting
A=1−
2β
p
(0 ≤ β < p)
and
B = −1
in Theorem 1, we obtain the following result (which also leads to the result of
zp
Owa and Patel [9, Corollary 3] when µ = 0, α = 1, β = 0 and g(z) = 1−z
):
J. K. Prajapat & R. K. Raina
7
Corollary 1. If f (z) ∈ Jp∗ (g; α, β), then
< (Qµn (f ∗ g)(z)) ≥ ρ∗n (r) > ρ∗n (1)
(n ∈ N0 ; |z| = r < 1),
where
0<
ρ∗n (r)
n
= 1 − 2(p − β)(µ + 1)
∞
X
(−1)k−1 rk
< 1.
(p + αk) (k + µ + 1)n
k=1
The result is sharp.
Furthermore on putting
α=n=1
and
g(z) =
zp
1−z
in Theorem 1, we get
Corollary 2. If f (z) ∈ Ap such that
1 + Az
f 0 (z)
≺p
p−1
z
1 + Bz
then
<
µ+1
z µ+1
Z
z
tµ−p f (t) dt
(z ∈ U),
≥ρ
(z ∈ U),
0
where
ρ = 1 + (B − A)(µ + 1)p
∞
X
k=1
(B)k−1 rk
.
(p + k) (k + µ + 1)
The result is sharp.
Theorem 2. If −1 ≤ Bj < Aj ≤ 1 (j = 1, 2), and fj (z) ∈ Jp (g; α, Aj , Bj )
(j = 1, 2), then
Ξ(z) ∈ Jp (g; α, 1 − 2η0 , −1),
(3.8)
where
Ξ(z) = ((f1 ∗ f2 ) ∗ g)(z)
(3.9)
and
η0 = 1 +
2(A1 − B1 )(A2 − B2 )
(1 − B1 )(1 − B2 )
p
1
F
1,
1;
+
1;
−
2
.
2 1
α
2
The result is sharp for B1 = B2 = −1.
Proof. Since fj (z) ∈ Jp (g; α, Aj , Bj ), it follows that
φi (z) = (1 − α)
(fi ∗ g)(z) α (fi ∗ g)0 (z)
+
,
zp
p
z p−1
and we can write that
φi (z) ∈ P(γi )
(3.10)
8
Some applications of differential subordination
for γi = (1 − Ai )/(1 − Bi ) (i = 1, 2). Integration by parts shows that
Z
p p− p z p −1
(fi ∗ g)(z) =
t α φi (t) dt
(i = 1, 2),
(3.11)
z α
α
0
and from (3.11), we can express
p p− p
(Ξ ∗ g)(z) =
z α
α
Z
z
p
t α −1 φ0 (t) dt,
(3.12)
0
where (for convenience sake)
φ0 (z) = (1 − α)
=
p −p
z α
α
(Ξ ∗ g)(z) α (Ξ ∗ g)0 (z)
+
zp
p
z p−1
Z z
p
t α −1 (φ1 ∗ φ2 )(t) dt.
(3.13)
0
Since φi (z) ∈ P(γi ) (i = 1, 2), it follows from Lemma 4 that
(φ1 ∗ φ2 )(z) ∈ P(γ3 ) for γ3 = 1 − 2(1 − γ1 )(1 − γ2 ).
(3.14)
Using the substitution t = uz in (3.13), we obtain the following equality:
<{φ0 (z)} =
p
α
Z
1
p
u α −1 <{(φ1 ∗ φ2 )(uz)} du.
0
In view of Lemma 3, the above last equality gives
Z
p 1 p −1
2(1 − γ3 )
α
u
du
<{φ0 (z)} ≥
2γ3 − 1 +
α 0
1 + u|z|
1
2(1 − γ3 )
u
2γ3 − 1 +
du.
1+u
0
2(A1 − B1 )(A2 − B2 )
p
1
= 1+
F
1,
1;
+
1;
−
2
.
2 1
(1 − B1 )(1 − B2 )
α
2
= η0 ,
p
>
α
Z
p
α −1
and this establishes (3.8).
Now, to prove the sharpness of the result, we find that (3.11) for B1 = B2 =
−1, gives
Z
(fi ∗ g)(z)
p − p z p −1 1 + Ai t
α
z
=
tα
dt
(i = 1, 2),
zp
α
1−t
0
and noting that
0
1 + A1 t
1 + A2 t
(1 + A1 )(1 + A2 )
∗
= 1 − (1 + A1 )(1 + A2 ) +
,
1−t
1−t
1−t
J. K. Prajapat & R. K. Raina
9
it then follows that
Z 1
p
p
(1 + A1 )(1 + A2 )
u α −1 1 − (1 + A1 )(1 + A2 ) +
φ0 (z) =
du.
α 0
1 − uz
If we choose z on the real axis, and let z → 1− , we obtain
(A1 + 1)(A2 + 1)
p
1
φ0 (z) → 1 +
+ 1;
−2 ,
2 F1 1, 1;
2
α
2
which proves Theorem 2.
Theorem 3. If fi (z) ∈ Ap (j = 1, 2) such that
(1 − α)
(fi ∗ g)(z) α (fi ∗ g)0 (z)
+
∈ P(ηi )
zp
p
z p−1
(0 ≤ ηi < 1; i = 1, 2), (3.15)
and the function Ξ is defined by (3.9), then
z (Ξ ∗ g)0 (z)
p
<
> p− ,
(Ξ ∗ g)(z)
α
(3.16)
provided that
(1 − η1 )(1 − η2 ) <
α + 2p
.
2
2 α 2 F1 1, 1; αp + 1; 12 − 2 + 2p
(3.17)
Proof. By hypothesis and Lemma 4, it follows that
"
0 #
(Ξ ∗ g)(z) α (Ξ ∗ g)0 (z)
(Ξ ∗ g)(z) α (Ξ ∗ g)0 (z)
α
< (1 − α)
+
+
+ z (1 − α)
zp
p
z p−1
p
zp
p
z p−1
(f1 ∗ g)(z) α (f1 ∗ g)0 (z)
(f2 ∗ g)(z) α (f2 ∗ g)0 (z)
+
+
= < (1 − α)
∗ (1 − α)
zp
p
z p−1
zp
p
z p−1
> 1 − 2(1 − η1 )(1 − η2 ),
(3.18)
which in view of Lemma 2 (for η = αp and β = 1 − 2(1 − η1 )(1 − η2 )) yields
(Ξ ∗ g)(z) α (Ξ ∗ g)0 (z)
p
1
< (1 − α)
+
> 1+2(1−η1 )(1−η2 ) 2 F1 1, 1; + 1;
−2 .
zp
p
z p−1
α
2
(3.19)
Again from (3.19) and Lemma 2, we have
2
p
1
(Ξ ∗ g)(z)
>
1
−
2(1
−
η
)(1
−
η
)
F
1,
1;
+
1;
−
2
.
<
1
2
2 1
zp
α
2
Now let
p(z) =
α z (Ξ ∗ g)0 (z)
−α+1
p (Ξ ∗ g)(z)
and
q(z) =
(Ξ ∗ g)(z)
.
zp
10
Some applications of differential subordination
Elementary computations shows that
0
α
(Ξ ∗ g)(z) α (Ξ ∗ g)0 (z)
(Ξ ∗ g)(z) α (Ξ ∗ g)0 (z)
+
+ z (1 − α)
+
(1 − α)
zp
p
z p−1
p
zp
p
z p−1
= q(z)
α
0
p (z) + z p (z)
p
2
= ψ(p(z), zp0 (z); z),
(3.20)
where ψ(u, v; z) = q(z) u2 + αp v . Thus, by using (3.18) in (3.20), we get
< (ψ(p(z), zp0 (z); z)) > 1 − 2(1 − η1 )(1 − η2 )
(z ∈ U),
and for all x, y ≤ − 21 (1 + x2 ), we infer that
<{ψ(ix, y, z)} = <(q(z))
≤
≤
α
y − x2
p
1
<(q(z))(α + (α + 2p)x2 )
2p
α
−
<(q(z)) ≤ 1 − 2(1 − η1 )(1 − η2 )
2p
(z ∈ U).
Now applying Lemma 5, we get <(p(z)) > 0 in U which implies that
z (Ξ ∗ g)0 (z)
p
<
> p−
(z ∈ U),
(Ξ ∗ g)(z)
α
and this complete the proof of Theorem 3.
Setting
α=1
and
g(z) =
zp
1−z
in Theorem 3, we get the following Corollary which in turn yields the
corresponding result of Lashin [4, Theorem 1] (for p = 1).
Corollary 3. If fi (z) ∈ Ap (i = 1, 2) and the function fi0 (z)/pz p−1 ∈
P(ηi ) (0 ≤ ηi < 1 (i = 1, 2)), then
z (f1 ∗ f2 )0 (z)
<
> 0
(z ∈ U),
(f1 ∗ f2 )(z)
provided that
(1 − η1 )(1 − η2 ) <
2p + 1
.
2
2 α 2 F1 1, 1; p + 1; 12 − 2 + 2p
J. K. Prajapat & R. K. Raina
4
11
Applications and observations
In view of the subordination (1.5) which is expressed in terms of the convolution
of the function (1.3) with the function defined by (1.1) (involving the arbitrary
sequence bk ), and appropriately selecting the sequence bk , the results presented
in this paper would lead further to various new or known results. For example, if
the sequence bk in (1.3) and the value of parameter α in (1.5) are, respectively,
chosen as follows:
δp
Γ(k + 1) Γ(p + 1 − λ)
and α =
(−∞ < λ < p + 1; δ ≥ 0; p ∈ N),
Γ(p + 1) Γ(k + 1 − λ)
p−λ
(4.1)
and in the process making use of the identity [12, p.112, Eq. (1.10)]:
bk =
z (Ω(λ,p)
f (z))0 = (p − λ) Ω(λ+1,p)
f (z) + λ Ω(λ,p)
f (z)
z
z
z
(z ∈ U)
(4.2)
in (1.3), then Theorems 2 and 3 correspond to the results given recently by Patel
(λ,p)
and Mishra [12, p. 116, Theorems 1.11 and 1.12]. The operator Ωz
used in
(4.2) is the extended fractional differintegral operator of order λ (−∞ < λ <
p + 1) ([12]) defined by
Ω(λ,p)
f (z) = z p +
z
=
∞
X
Γ(n + 1) Γ(p + 1 − λ)
an z p
Γ(p
+
1)Γ(n
+
1
−
λ)
n=p+1
Γ(p + 1 − λ) λ λ
z Dz f (z)
Γ(p + 1)
(z ∈ U),
where Dzλ represents, respectively, the fractional integral of f (z) of order
−λ (−∞ < λ < 0) , and the fractional derivative of f (z) of order λ when
(0 < λ < p + 1) (Owa [8]).
On the other hand, if we set the coefficient bk in (1.3) and the value of
parameter α in (1.5), respectively, as follows:
bk =
(λ + p)k−p
(k − p)!
and
α=
δp
λ+p
(λ > −p; δ > 0; p ∈ N),
(4.3)
and in the process use the identity [3, p. 124, Eq.(4)]:
0
z Dλ+p−1 f (z) = (λ+p) Dλ+p f (z)−λ Dλ+p−1 f (z)
(λ > −p; p ∈ N; z ∈ U)
(4.4)
in (1.3), then Theorem 2 corresponds to the known result of Dinggong and Liu
[3, p. 129, Theorem 4]. The operator Dλ+p−1 used in (4.4) is the generalized
Ruscheweyh derivative which is defined by ([3, p. 123, Eq. (1)])
Dλ+p−1 f (z) =
zp
∗ f (z)
(1 − z)λ+p
(λ > −p; f ∈ Ap ).
12
Some applications of differential subordination
Furthermore, if we choose the coefficients bk in (1.3) and the value of the
parameter α in (1.5), respectively, as follows:
σ
λ
p+1
and
α=
bk =
(σ > 0; λ ≥ 0; p ∈ N), (4.5)
k+1
(p + 1)
and apply the following identity([10, p. 3, Eq. (1.17)]):
0
z (I σ f (z)) = (p + 1)I σ−1 f (z) − I σ f (z)
(p ∈ N; σ > 0)
(4.6)
in (1.3), then the result contained in Theorem 2 also yields a recently established
result due to Ozkan [10, p. 4, Theorem 2.4], where the operator I σ f (z) in (4.6)
is defined by ([10])
Z
z σ−1
(p + 1)σ z I σ f (z) =
log
f (t)dt
z Γ(σ) 0
t
σ
∞ X
p+1
p
ak z k
(f ∈ Ap ; p ∈ N; σ > 0).
=z +
k+1
k=p+1
We conclude this paper by observing that on specializing the coefficients bk
in (1.3) and the parameter α in (1.5) suitably, one may also obtain the results
given by Ding et al. [2], Lashian [4], Obradovic [6], Owa [7], Owa and Patel [9],
Patel et al. [11], Sham et al. [15] and Srivastava et al. ([18], [19]).
Acknowledgements. The authors express their sincerest thanks to the referees for some useful suggestions.
References
[1] M. Abramowitz and I. A. Stegun (Editors), Handbook of Mathematical
Functions and Formulas, Graphs and Mathematical Tables, Dover Publications, New York, 1971.
[2] S. Ding, Y. Ling and G. Bao, Some properties of a class of analytic functions, J. Math. Anal. Appl., 195, (1995), 71-81.
[3] Y. Dinggong and J. L. Liu, On a class of analytic functions involving
Ruscheweyh derivatives, Bull. Korean Math. Soc., 39(1), (2002), 123-131.
[4] A. Y. Lashin, Some convolution properties of analytic functions, Appl.
Math. Lett. 18, (2005), 135-138.
[5] S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics,
Vol. 225, Marcel Dekker, New York, 2000.
[6] M. Obradovic, On certain inequalities for some regular functions in |z|,
Int. J. Math. & Math. Sci., 8, (1985), 671-681.
J. K. Prajapat & R. K. Raina
13
[7] S. Owa, Certain integral operators, Proc. Japan Acad., 67, (1991), 88-93.
[8] S. Owa, On the distortion theorems I, Kyungpook Math. J., 18, (1978),
53-59.
[9] S. Owa and J. Patel, Properties of certain integral operators, Southeast
Asian Bull. Math., 24, (2000), 411-419.
[10] O. Ozkan, Some subordination results on multivalent functions defined
by integral operator, J. Inequal. Appl., Volume 2007, (2007), Article ID
71616, 1-8 (electronic).
[11] J. Patel, N. E. Cho and H. M. Srivastava, Certain subclasses of multivalent function associated with a family of linear operators, Math. Comput.
Modelling, 43, (2006), 320-328.
[12] J. Patel and A. K. Mishra, On certain subclasses of multivalent function associated with an extended fractional differintegral operator, J. Math. Anal.
Appl., 332, (2007),109-122.
[13] D. Z. Pashkouleva, The starlikeness and spiral-convexity of certain subclasses of analytic functions, in: H. M. Srivastava and S. Owa (Editors),
Current Topics in Anaytic Function Theory, pp. 266-273, World Scientific
Publishing Company, Singapore, New Jersey, London and Hongkong, 1992.
[14] S. Poonusamy, Differential subordination and Bazilevic functions, Proc.
Indian Acad. Sci. (Math. Sci.), 105(2), (1995), 169-186.
[15] S. Sham, S. R. Kulkurni and J. M. Jahangiri, Subordination properties of
p-valent functions defined by integral operators, Internat. J. Math. Math.
Sci., Volume 2006, (2006), Article ID 94572, 1-3 (electronic).
[16] H. M. Srivastava and S. Owa (Eds.), Univalent functions, Fractional calculus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto,
1989.
[17] H. M. Srivastava and J. Patel, Applications of differential subordination
to certain subclasses of meromorphically multivalent functions, J. Inequal.
Pure Appl. Math., 6(3), (2005), Article 88, 15 pp.
[18] H. M. Srivastava, J. Patel and G. P. Mohapatra, Some applications of
differential subordination to a general class of multivalent functions, Adv.
Stud. Contemp. Math., 3(1), (2001), 1-15.
[19] H. M. Srivastava, J. Patel and G. P. Mohapatra, A certain class of p-valently
analytic functions, Math. Comput. Modelling, 41, (2005), 321-324.
[20] J. Stankiewicz and Z. Stankiewicz, Some applications of Hadamard convolution in the theory of functions, Ann. Univ. Mariae Curie-Sklodowska
Sect., A 40, (1986), 251-265.
14
Some applications of differential subordination
J. K. Prajapat
Departament of Mathematics
Bhartiya Institute of Engineering and Technology
Near Sanwali By-Pass Circle
Sikar-332021, Rajasthan, India.
email: jkp 0007@rediffmail.com
R. K. Raina
10/11 Ganpati Vihar, Opposite Sector 5,
Udaipur 313002, Rajasthan, India
email: rkraina 7@hotmail.com