MATEMATIQKI VESNIK
originalni nauqni rad
research paper
64, 1 (2012), 33–38
March 2012
INEQUALITIES INVOLVING A CLASS OF ANALYTIC FUNCTIONS
Rabha W. Ibrahim
Abstract. In the present paper, we define a new class of normalized analytic functions based
on the Φ-like type. Some inequalities are introduced based on the concept of the subordination
in the unit disk. The convexity techniques are used to obtain the main result. Some applications
are posed on classes of functions such as starlike functions, convex functions, starlike and convex
functions of complex order.
1. Introduction
Let H be the class of functions analytic in the unit disk U = {z : |z| < 1}
and for a ∈ C (set of complex numbers) and n ∈ N (set of natural numbers), let
H[a, n] be the subclass of H consisting of functions of the form f (z) = a + an z n +
an+1 z n+1 + · · · . Let A be the class of functions f , analytic in U and normalized
by the conditions f (0) = f 0 (0) − 1 = 0. A function f ∈ A is called starlike of order
µ if it satisfies the following inequality
n zf 0 (z) o
<
> µ, (z ∈ U )
f (z)
for some 0 ≤ µ < 1. We denoted this class S(µ). A function f ∈ A is called convex
of order µ if it satisfies the following inequality
n zf 00 (z)
o
<
+
1
> µ, (z ∈ U )
f 0 (z)
for some 0 ≤ µ < 1. We denoted this class C(µ). We note that f ∈ C(µ) if and only
if zf 0 ∈ S(µ).
Let f be analytic in U , g analytic and univalent in U and f (0) = g(0). Then,
by the symbol f (z) ≺ g(z) (f subordinate to g) in U , we shall mean f (U ) ⊂ g(U ).
Let φ : C2 → C and let h be univalent in U . If p is analytic in U and satisfies
the differential subordination φ(p(z)), zp0 (z)) ≺ h(z) then p is called a solution of
2010 AMS Subject Classification: 30C99, 30C45.
Keywords and phrases: Univalent functions; Starlike functions; Convex functions; Differential subordination; Superordination; Unit disk; Starlike and convex functions of complex order;
Φ-like functions; Convexity techniques.
33
34
R. W. Ibrahim
the differential subordination. The univalent function q is called a dominant of
the solutions of the differential subordination, p ≺ q. If p and φ(p(z)), zp0 (z)) are
univalent in U and satisfy the differential superordination h(z) ≺ φ(p(z)), zp0 (z))
then p is called a solution of the differential superordination. An analytic function
q is called subordinant of the solution of the differential superordination if q ≺ p.
We use the notation s(g) = {f ∈ H(U ) : f ≺ g}, for details (see [1]).
The function f ∈ A is called Φ-like if
n zf 0 (z) o
> 0, z ∈ U.
<
Φ(f (z))
This concept was introduced by Brickman [2] and established that a function f ∈ A
is univalent if and only if f is Φ-like for some Φ.
Definition 1. Let Φ be analytic function in a domain containing f (U ), Φ(0) =
0, Φ0 (0) = 1 and Φ(ω) 6= 0 for ω ∈ f (U ) − 0. Let q(z) be a fixed analytic function
in U , q(0) = 1. The function f ∈ A is called Φ-like with respect to q if
zf 0 (z)
≺ q(z), z ∈ U.
Φ(f (z))
Ruscheweyh [3] investigated this general class of Φ-like functions.
Let X be a locally convex linear topological space. For a subset U ⊂ X
the closed convex hull of U is defined as the intersection of all closed convex sets
containing U and will be denoted by co(U ). If U ⊂ V ⊂ X then U is called an
extremal subset of V provided that whenever u = tx + (1 − t)y where u ∈ U ,
x, y ∈ V and t ∈ (0, 1) then x, y ∈ U . An extremal subset of U consisting of one
point is called an extreme point of U . The set of the extreme points of U will be
denoted by E(U ).
Remark 1. If L : H(U ) → H(U ) is an invertible linear map and F ⊂ H(U ) is
a compact subset, then L(co(F )) = co(L(F )) and the set E(co(F )) is in one-to-one
correspondence with EL(co(F )).
In the present paper, we estimate the upper and lower bound for functions
f ∈ A in the class Apm (α1 , . . . , αp ; Φ1 , . . . , Φp ), p ≥ 1, m ∈ N which satisfy the
condition
np
o
zf 0 (z)
zf 0 (z)
zf 0 (z)
<
[m]1 + α1
+ α2
+ · · · + αp
− (α1 + · · · + αp ) > 0,
Φ1 (f (z))
Φ2 (f (z))
Φp (f (z))
(1)
where αj , j = 1, . . . , p, are complex numbers and Φj satisfies Definition 1. In order
to obtain our results, we need the following technical lemmas.
Lemma 1. [4] Suppose that Fα is defined by the equality
³ 1 + cz ´α
Fα (z) =
, (|c| ≤ 1, c 6= −1).
1−z
Inequalities involving a class of analytic functions
35
If α ≥ 1 then co(s(F α )) consists of all functions in H(U ) represented by
Z 2π ³
1 + cze−it ´α
f (z) =
dµ(t),
1 − ze−it
0
where µ is a positive measure on [0, 2π] having the property µ([0, 2π]) = 1 and
E(co(s(F α ))) =
o
n 1 + cze−it ¯
¯
t
∈
[0,
2π]
.
¯
1 − ze−it
Lemma 2. [4] If J : H(U ) → R is a real-valued, continuous convex functional
and F is a compact subset of H(U ), then
max{J(f ) : f ∈ co(F)} = max{J(f ) : f ∈ F} = max{J(f ) : f ∈ E(co(F))}.
In the particular case if J is a linear map then we also have:
min{J(f ) : f ∈ co(F)} = min{J(f ) : f ∈ F} = min{J(f ) : f ∈ E(co(F))}.
Lemma 3. [5] For z ∈ U we have
nP
j
<
zn o
1
> − , (z ∈ U ).
3
n=1 n + 2
2. The main result
Our main result is the following:
Theorem 1. Let f ∈ Apm (α1 , . . . , αp ; Φ1 , . . . , Φp ), p ≥ 1, m ∈ N and αj ,
j = 1, . . . , p, are complex numbers. Then
Pm
k m−1
³P
´
³
h zf 0 (z)
i´
∞
k=0 Cm Cm+n−k−1 n
1 + inf <
z < < 1 + αp
−1
z∈U
Q(n)
Φp (f (z))
n=1
Pm
m−1
k
³P
´
∞
k=0 Cm Cm+n−k−1 n
< 1 + sup <
z , (2)
Q(n)
z∈U
n=1
where
Q(u) := 1 + α1 u + α2 u(u − 1) + · · · + αp u(u − 1) · . . . · (u − p + 1).
Proof. Since f ∈ Apm (α1 , . . . , αp ; Φ1 , . . . , Φp ), we have
p
zf 0 (z)
zf 0 (z)
zf 0 (z)
1+z
[m]1 + α1
+ α2
+ · · · + αp
− (α1 + · · · + αp ) ≺
Φ1 (f (z))
Φ2 (f (z))
Φp (f (z))
1−z
36
R. W. Ibrahim
which is equivalent to
1 + α1
³ 1 + z ´m
zf 0 (z)
zf 0 (z)
zf 0 (z)
+ α2
+ · · · + αp
− (α1 + · · · + αp ) ≺
.
Φ1 (f (z))
Φ2 (f (z))
Φp (f (z))
1−z
In view of Lemma 1,
1 + α1
zf 0 (z)
zf 0 (z)
zf 0 (z)
+ α2
+ · · · + αp
− (α1 + · · · + αp )
Φ1 (f (z))
Φ2 (f (z))
Φp (f (z))
Z 2π ³
1 + ze−it ´α
=
dµ(t) := h(z),
1 − ze−it
0
(3)
where µ is a positive measure on [0, 2π] having the property µ([0, 2π]) = 1. Assume
that
h zf 0 (z)
i
∞
P
1 + αp
−1 =1+
bn z n , (z ∈ U ).
(4)
Φp (f (z))
n=1
Then
1 + α1
zf 0 (z)
zf 0 (z)
zf 0 (z)
+ α2
+ · · · + αp
− (α1 + · · · + αp )
Φ1 (f (z))
Φ2 (f (z))
Φp (f (z))
∞
P
=1+
bn Q(n)z n . (5)
n=1
On the other hand, we observe that
Z 2π ³
i Z 2π
∞ h P
n
P
1 + ze−it ´α
m−1
n
k
C
e−int dµ(t)
dµ(t)
=
1
+
C
m m+n−k−1 z
1 − ze−it
n=1 k=0
0
0
(6)
such that Cji = 0, i > j and
n
P
k=0
k m−1
Cm
Cm+n−k−1 =
m
P
k=0
m−1
k
Cm
Cm+n−k−1
.
(7)
Thus (6) and (7) imply
1+
∞
P
n=1
n
bn Q(n)z = 1 +
and
bn =
∞ h P
m
P
n=1
k=0
m−1
k
Cm+n−k−1
Cm
´
m
1 ³P
m−1
k
Cm
Cm+n−k−1
Q(n) k=0
Z
2π
i Z
zn
2π
e−int dµ(t)
0
e−int dµ(t).
0
Consequently,
´
∞
m
P
zf 0 (z)
1 ³P
k m−1
Cm
Cm+n−k−1 z n
1 + αp [
− 1] = 1 +
Φp (f (z))
n=1 Q(n) k=0
If
Z
n
H = h ∈ H(U ) : h(z) =
0
2π
³ 1 + ze−it ´α
1−
ze−it
Z
2π
e−int dµ(t).
0
o
dµ(t), µ([0, 2π]) = 1
Inequalities involving a class of analytic functions
37
and
o
n
zf 0 (z)
Q = q ∈ H(U ) : q(z) = 1 + αp [
− 1], f ∈ Apm (α1 , . . . , αp ; Φ1 , . . . , Φp )
Φp (f (z))
then the correspondence L : H → Q, L(h) = q(z) defines an invertible linear map
and according to Remark 1, the extreme points of the class Q are
´
∞
m
P
1 ³P
k m−1
qt (z) = 1 +
Cm
Cm+n−k−1 z n e−int , (z ∈ U, t ∈ [0, 2π)).
n=1 Q(n) k=0
Hence in view of Lemma 2 this implies the assertion of Theorem 1.
3. Applications
Theorem 2. Let f be starlike in U . Then
n zf 0 (z) o
2
<<
< 2.
3
f (z)
0
(z)
Proof. Since f is starlike, then <{ zff (z)
} > 0 implies that f ∈ A11 (1, 0, . . . , 0; f ).
Thus according to Theorem 1, we obtain the following inequalities:
³P
´
n zf 0 (z) o
³P
´
∞
∞
1
1
1 + inf <
zn ≤ <
≤ 1 + sup <
zn .
z∈U
f (z)
z∈U
n=1 n + 1
n=1 n + 1
In virtue of Lemma 3 it yields
³P
∞
1 + inf <
z∈U
´
³P
´
j
1
1
z n ≥ 1 + inf <
zn
z∈U
n=1 n + 1
n=1 n + 1
³P
´ 2
j
1
≥ 1 + inf <
zn > .
z∈U
3
n=1 n + 2
On the other hand, and by residue theory, we pose
³P
´
³P
´
³ z ´
∞
∞
1
1 + sup <
z n < 1 + supz∈U <
z n = 1 + supz∈U <
= 2.
1−z
z∈U
n=1 n + 1
n=1
Hence the proof.
Theorem 3. Assume that f ∈ A and satisfies
³ zf 0 (z) ´ 1
<
> .
f (z)
2
Then
³ zf 0 (z) ´ e2π (1 + π) − (1 − π)
e2π + 2πeπ − 1
≤
1
+
<
≤
.
e2π − 1
f (z)
e2π − 1
The result is sharp.
(8)
38
R. W. Ibrahim
Proof. Condition (8) implies that f ∈ A21 (1, 1, 0, . . . , 0; f, f ). Thus according
to Theorem 1, we have the following inequalities:
³P
´
n zf 0 (z) o
³P
´
∞
∞
2
2
n
n
1 + inf <
z
≤
<
≤
1
+
sup
<
z
.
2
2
z∈U
f (z)
z∈U
n=1 n + 1
n=1 n + 1
The minimum and maximum principle for harmonic functions imply that
³P
´
³P
´
∞
∞
2
2
n
int
1 + inf <
z
=
1
+
inf
<
e
2
2
z∈U
t∈[0,2π]
n=1 n + 1
n=1 n + 1
and
³P
∞
1 + sup <
´
³P
´
∞
2
2
n
int
z
=
1
+
sup
<
e
.
2
2
n=1 n + 1
n=1 n + 1
t∈[0,2π]
z∈U
Again by using the residue theory we deduce that
1+<
³P
∞
´ π(et + e2π−t )
2
int
e
=
,
2
e2π − 1
n=1 n + 1
(t ∈ [0, 2π])
consequently, we obtain the desired assertion.
In the same manner we have the following result:
Theorem 4. Assume that f ∈ A11 ( 1b , 0 . . . , 0; f ),
b ∈ C\{0}. Then
o
∞ (−1)k <(b)
P
1 n zf 0 (z)
1+ <
−1 ≥1+
.
b
f (z)
k=1 <(b) + k
The result is sharp.
REFERENCES
[1] S.S. Miller, P.T. Mocanu, Differential Subordinantions: Theory and Applications, Pure and
Applied Mathematics No.225 Dekker, New York, 2000.
[2] L. Brickman, Φ-like analytic functions, I, Bull. Amer. Math. Soc. 79 (1973), 555–558.
[3] St. Ruscheweyh, A subordination theorem for Φ-like functions, J. London Math. Soc. 13
(1976), 275–280.
[4] D.J. Hallenbeck, T.H. Mac Gregor, Linear Problems and Convexity Techniques in Geometric
Function Theory, Pitman Advanced Publishing Program, 1984.
[5] J.M. Jahangiri, K. Farahmand, Partial sums of functions of bounded turning, Inter. J. Math.
Math. Sci. 1 (2004), 45–47.
(received 13.09.2010; available online 20.01.2011)
School of Mathematical Sciences, Faculty of Sciences and Technology, UKM, Malaysia
E-mail: rabhaibrahim@yahoo.com