2_100137_altintas.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 1, 2012, 16-24
ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA

29 JUNE - 02 JULY 2011, ISTANBUL TURKEY

Majorization for Certain Analytic Functions

Osman ALTINTAS

Department of Mathematics, Faculty of Education, Başkent University, Ankara, Turkey

Abstract. In this paper two subclasses Sδ
p,q

�

γ,A, B
�

and Cδ
p,q

�

γ,A, B
�

of p-valently starlike and p-
valently convex functions of complex order γ 6= 0 in the open unit disk U are introduced and for
these classes several majorization problems are discussed.

2000 Mathematics Subject Classifications: 30C45
Key Words and Phrases: Analytic function, p-valent function, Starlike function, Convex function,
Majorization problems, Fractional derivative.

1. Introduction and Definitions

Definition 1 ([see 5]). Let the functions f (z) and g(z) be analytic in the open unit disk

U = {z : z ∈ C and | z |< 1} .
We say that f (z) is majorized by g(z) and write

f (z)≪ g(z) (1)

if there exists a function φ(z) analytic in ∪, such that

| φ(z) |≤ 1 and f (z) = φ(z)g(z). (2)

Also, we say that f (z) is subordinate to g(z) and write

f (z)≺ g(z)

if there exist a function w(z) analytic in U , such that

w (0) = 0, |w (z)| ≤ |z| and f (z) = g (w (z)) .

Email address: oaltintas�baskent.edu.tr (O. Altıntaş)

http://www.ejpam.com 16 c© 2012 EJPAM All rights reserved.



O. Altıntaş / Eur. J. Pure Appl. Math, 5 (2012), 16-24 17

Definition 2 ([see 8]). The fractional derivative of order δ is defined by

Dδz f (z) =
1

Γ(1−δ)
d

dz

z
∫

0

f (ζ)

(z − ζ)δ
dζ (0¶ δ < 1) (3)

where f (z) is an analytic function in a simply connected region of the z−plane containing the

origin and the multiplicity of (z − ζ)−δ is removed by requiring log (z − ζ) to be real when

z − ζ > 0.

Definition 3 ([see 8]). Under the hypotheses of definition 2, the fractional derivative of order

(n+ δ) is defined by

Dn+δ
z f (z) =

d

dzn
Dδz f (z) . (4)

Several majorization problems investigated by Altıntaş and Owa [1], Altıntaş et al. [2]
and [3].

Let Ap denote the class of functions f normalized by

f (z) = zp +

∞
∑

n=p+1

anzn (P ∈ N = {1,2,3, . . .})

which are analytic and p − valent in U . Also let a function f ∈ Ap is said to be in the class

Sδp,q

�

γ,A, B
�

if and only if

1+
1

γ

 

z f (q+δ+1)(z)

f (q+δ)(z)
− p+ q+ δ

!

≺ 1+ AZ

1+ BZ
(5)

where γ ∈ C \ {0}, p ∈ N, q ∈ N0 = N∪ {0}, 0¶ δ < 1, −1¶ B < A¶ 1 and
�

�γ(A− B) + (p− q− δ)B
�

� ¶
�

�p− q− δ
�

� .

Furthermore a function f ∈ Ap is said to be in the class Cδp,q

�

γ,A, B
�

if and only if

1+
1

γ

 

1+
z f (q+δ+2)(z)

f (q+δ+1)(z)
− p+ q+ δ

!

≺ 1+ AZ

1+ BZ
(6)

where γ ∈ C \ {0}, p ∈ N, q ∈ N0, 0¶ δ < 1, −1¶ B < A¶ 1 and
�

�γ(A− B) + (p− q− δ)B
�

� ¶
�

�p− q− δ
�

� .

We have the following relationships (from [3, 11, 2], respectively)

S0
p,q(γ, 1,−1) = Sp,q(γ).

C0
p,q(γ, 1,−1) = Cp,q(γ).



O. Altıntaş / Eur. J. Pure Appl. Math, 5 (2012), 16-24 18

S0
p,0(γ, 1,−1) = S(γ) and C0

p,0(γ, 1,−1) = C(γ).

S(γ) and C(γ) were considered by Nasr and Aouf in [6].

S0
p,0(1−α, 1,−1) = S∗(α) and C0

p,0(1−α, 1,−1) = C(α)

denote respectively the class of starlike and convex functions of order α, (0 ¶ α < 1) which
were introduced by Robertson in [9].

2. Majorization Problems for the Class Sδ
p,q(γ, A, B)

We begin by proving.

Theorem 1. Let the function f (z) be in the class Ap and suppose that g ∈ Sδp,q(γ,A, B). If

f (q+δ)(z) is majorized by g(q+δ)(z) in U for q ∈ No and 0¶ δ < 1, then

�

�

� f (q+δ+1)(z)

�

�

� ¶

�

�

�g(q+δ+1)(z)

�

�

�

�|z| ¶ r1
�

(7)

where r1 = r1(p,q,δ,γ,A, B) is the smallest positive root of the equation

�

�γ(A− B) + (p− q− δ)B
�

� r3 − (p− q− δ+ 2 |B|)r2− [
�

�γ(A− B) + (p− q− δ)B
�

�

+ 2]r + p− q− δ = 0 (8)

where p ∈ N, q ∈ N0, γ ∈ C \ {0}, 0¶ δ < 1 and

�

�γ(A− B) + (p− q− δ)B
�

� ¶
�

�p− q− δ
�

� .

Proof. Since g ∈ Sδp,q(γ,A, B), we obtain from (5)

1+
1

γ

 

z g(q+δ+1)(z)

g(q+δ)(z)
− p+ q+ δ

!

=
1+ Aω(z)

1+ Bω(z)
(9)

where
ω(0) = 0 and |ω(z)| ¶ |z| (z ∈ U). (10)

From (9) we readily obtain

z g(q+δ+1)(z)

g(q+δ)(z)
=

p− q− δ+ �γ(A− B) + (p− q− δ)B�ω(z)
1+ Bω(z)

. (11)

Using (10) in (11) we find

�

�

�g(q+δ)(z)

�

�

� ¶
(1+ |B| |z|) |z|

p− q− δ−
�

�γ(A− B) + (p− q− δ)B
�

� |z|
�

�

�g(q+δ+1)(z)

�

�

� . (12)



O. Altıntaş / Eur. J. Pure Appl. Math, 5 (2012), 16-24 19

Since f (q+δ)(z) is majorized by g(q+δ)(z) from (2) we have

f (q+δ+1)(z) = φ(z)g(q+δ+1)(z) +φ′(z)g(q+δ)(z), (13)

φ(z) is satisfies the inequality [cf. Nehari 7, p. 168]:

�

�φ′(z)
�

� ¶
1−

�

�φ(z)
�

�

2

1− |z|2 (z ∈ U) (14)

and using (12) and (14) in (13), we get

�

�

� f (q+δ+1)(z)

�

�

� ¶
�

�φ(z)
�

�+
1−

�

�φ(z)
�

�

2

1− |z|2
(1+ |B| |z|) |z|

p− q− δ−
�

�γ(A− B) + (p− q− δ)B
�

� |z|
�

�

�g(q+δ+1)(z)

�

�

�

(15)
which, upon setting

|z| = r,
�

�φ(z)
�

� = ρ (0¶ ρ ¶ 1)

leads us to the inequality

�

�

� f (q+δ+1)(z)

�

�

� ¶
θ(ρ)

(1− r2)
�

p− q− δ−
�

�γ(A− B) + (p− q− δ)B
�

� r
� g(q+δ+1)(z) (16)

where

θ(ρ) = −(r+ |B| r2)ρ2+(1− r2)p−q−δ−
�

�γ(A− B) + (p− q−δ)B
�

� r]ρ+(r+ |B| r2) (17)

takes on its maximum value at ρ = 1 with r = r1(p,q,δ,γ,A, B) gives by (8) if

0¶ σ ¶ r1(p,q,δ,γ,A, B)

then the function ∧(ρ) defined by

∧(ρ) =−(σ+σ2 |B|)ρ2+(1−σ2)
�

p− q− δ−
�

�γ(A− B) + (p− q− δ)B
�

�σ
�

ρ+(σ+σ2 |B|)
(18)

is an increasing function on the interval 0¶ ρ ¶ 1 so that

∧(ρ)¶ ∧(1) = (1−σ2)
�

p− q− δ−
�

�γ(A− B) + (p− q− δ)B
�

�σ
�

(0¶ ρ ¶ 1; 0¶ σ ¶ r1(p,q,δ,γ,A, B)).

Hence, by setting ρ = 1 in (16), we conclude that Theorem 1 holds true for |z| ¶ r1(p,q,δ,γ,A, B)

is given by (8). This completes the proof of Theorem 1.

Corollary 1 ([see 3]). Let the function f (z) be in the class Ap and suppose that

g ∈ S0
p,q(γ, 1,−1). If f (q)(z) is majorized by g(q)(z) in U, then

�

�

� f (q+1)(z)

�

�

�¶

�

�

�g(q+1)(z)

�

�

� (|z| ¶ R1)



O. Altıntaş / Eur. J. Pure Appl. Math, 5 (2012), 16-24 20

where

R1 = R1(p,q,δ) =
k−
Æ

k2 − 4(p− q)
�

�2γ− p+ q
�

�

2
�

�2γ− p+ q
�

�

(19)

(k = p− q+ 2+
�

�2γ− p+ q
�

� , p ∈ N,q ∈ N0,γ ∈ C \ {0}).
Proof. If we set δ = 0, A= 1, B = −1 in Theorem 1, then

�

�

� f (q+1)(z)

�

�

� ¶

�

�

�g(q+1)(z)

�

�

� |z| ¶ R1

where R1 = R1(p,q,δ) is the smallest positive root of the equation
�

�2γ− p+ q
�

� r3 − (p− q+ 2)r2− [
�

�2γ− p+ q
�

�+ 2]r + p− q = 0

r = −1 is the root of the above equation and we obtain
�

�2γ− p+ q
�

� r2 − (
�

�2γ− p+ q
�

�+ p− q+ 2)r + p− q = 0. (20)

and the positive root of the equation (20) is R1 = R1(p,q,δ).

Corollary 2 ([see 2]). Let the function f (z) be in the class A1 and suppose that g ∈ S0
1,0(γ, 1,−1).

If f (z) is majorized by g(z) in U, then

�

�

� f
′
(z)

�

�

�¶

�

�

�g
′
(z)

�

�

� (|z| ¶ R2)

where

R2 = R2(γ) =
3+

�

�2γ− 1
�

�−
Æ

9+ 2
�

�2γ− 1
�

�+
�

�2γ− 1
�

�

2

2
�

�2γ− 1
�

�

Corollary 3 ([see 5]). Let f (z) be in the class A1 and suppose that g(z) ∈ S0
1,0(1,1,−1). If f (z)

is majorized by g(z) in U, then

�

�

� f
′
(z)

�

�

� ¶
�

�g′(z)
�

� (|z| ¶ R3)

where R3 = 2−p3.

3. Majorization Problems for the Class Cδ
p,q(γ, A, B).

The proof Theorem 2 is based upon the following Lemmas.

Lemma 1 ([see 10, Theorem 1]). If f ∈ Cδp,q(γ,A, B) (γ ∈ C \ {0}) then

Re

�

1+
1

γ

�

z f (q+δ+2)(z)

f (q+δ+1)(z)
− p+ q+ δ+ 1

��

>
1− A

1− B
(21)



O. Altıntaş / Eur. J. Pure Appl. Math, 5 (2012), 16-24 21

Proof. If f ∈ Cδp,q(γ,A, B) then we have from (6)

1+
1

γ
(
z f (q+δ+2)(z)

f (q+δ+1)(z)
− p+ q+δ+ 1) =

1+ Aw(z)

1+ Bw(z)
(22)

where w(0) = 0 and |w(z)| ¶ |z|, (−1¶ B < A¶ 1). We let

h(z) =
1+Aw(z)

1+ Bw(z)
(23)

and

h(z) = u+ iv, |w(z)|2 =
�

�

�

�

h(z)− 1

A− Bh(z)

�

�

�

�

2

¶ 1

and
(1− B2)u2 − 2(1− AB)u+ 1− A2

¶ 0 (24)

from (24) implies that
1− A

1− B
¶ Reh(z) = u ¶

1+ A

1+ B
. (25)

The following lemma is proved in [3] for δ = 0.

Lemma 2. If f ∈ Cδp,q(γ,A, B) (γ ∈ C \ {0}) then f ∈ Sδp,q(
1
2
γ,A, B) that is

Cδp,q(γ,A, B) ⊂ Sδp,q(
γ

2
,A, B) (26)

Proof. We know that all convex function in U is starlike of order 1
2

in U, [see 4, p. 7] or,
equivalently

Re[1+
z f ′′(z)
f ′(z)

]> 0⇒ Re[
z f ′(z)
f (z)

]>
1

2
. (27)

If we let

Re[1+
z f ′′(z)
f ′(z)

]> α for f (z)−→ f (q+δ)(z),

and using Lemma 1, we have

Re[1+
z f (q+δ+2)(z)

f (q+δ+1)(z)
− p+ q+ δ+ 1)]>

1− A

1− B
(28)

or

Re[1+
1

1−α(
z f (q+δ+2)(z)

f (q+δ+1)(z)
− p+ q+ δ+ 1]> 0. (29)

This implies that

1+
1

1−α(
z f (q+δ+2)(z)

f (q+δ+1)(z)
− p+ q+ δ+ 1=

1−w(z)

1+w(z)
. (30)



O. Altıntaş / Eur. J. Pure Appl. Math, 5 (2012), 16-24 22

So, we have

1+
1

γ
(
z f (q+δ+2)(z)

f (q+δ+1)(z)
− (p+ q+ δ+ 1) =

γ+ (γ− 2+ 2α)w(z)

γ(1+w(z))
. (31)

On the other hand we know that

z f p(z)

f (z)
> α⇒ Re(1+

1

1−α
z f p(z)

f (z)
)> 0. (32)

Similarly using (27) and (29) we obtain the following relations.

[1+
1

1−α(
z f (q+δ+1)(z)

f (q+δ)(z)
− p+ q+ δ)]>

1

2
, (33)

1+
1

1−α(
z f (q+δ+1)(z)

f (q+δ)(z)
− p+ q+ δ) =

1

1+w(z)
, (34)

1+
2

γ
(
z f (q+δ+1)(z)

f (q+δ)(z)
− p+ q+ δ) =

γ+ (γ− 2+ 2α)

γ(1+w(z))
. (35)

(36)

The inclusion property (26) is easily seen that from (31) and (35).

Upon replacing γ in Theorem 1 by 1
2
γ, if we apply Lemma 2 we have,

Theorem 2. Let the function f (z) be in the class Ap and suppose that g(z) ∈ Cδp,q(γ,A, B). If

f (q+δ)(z) is majorized by g(q+δ)(z) ∈ U, for p ∈ N q ∈ N0 and 0¶ δ < 1 then

�

�

� f (q+δ+1)(z)

�

�

� ¶

�

�

�g(q+δ+1)(z)

�

�

� (|z| ≤ r2) (37)

where r2 = r2(p,q,δ,γ,A, B) is the smallest positive root of the equation

�

�

�

�

1

2
γ(A− B) + (p− q− δ)B

�

�

�

�

r3 − (p− q− δ+ 2 |B|)r2− [1
2
γ(A− B) + (p− q− δ) |B|

+ 2]r + p− q− δ = 0

where p ∈ N, q ∈ N0, γ ∈ C \ {0}, 0¶ δ < 1, and

�

�

�

�

1

2
γ(A− B) + (p− q− δ)B

�

�

�

�

¶
�

�p− q− δ
�

� .

Corollary 4 ([see 3]). Let the function f (z) be in the class Ap and suppose that

g(z) ∈ C0
p,q(γ, 1,−1). If f (q)(z) is majorized by g(q)(z) in U, then

�

�

� f (q+1)(z)

�

�

�¶

�

�

�g(q+1)(z)

�

�

� (|z| ≤ R1)



REFERENCES 23

where

R1 = R1(p,q,δ) =
µ−

Æ

µ2 − 4(p− q)
�

�γ− p+ q
�

�

2
�

�γ− p+ q
�

�

µ = 2+ p− q+
�

�γ− p+ q
�

� , p ∈ N,q ∈ N0,γ ∈ C \ {0}
Proof. We let δ = 0, A= 1, B = −1 in Theorem 2.

Corollary 5 ([see 2]). Let the function f (z) be in the class A1 and suppose that g(z) ∈ C0
1.0(1,1,−1).

If f (z) is majorized by g(z) in U, then

�

�

� f
′
(z)

�

�

�¶

�

�

�g
′
(z)

�

�

� (|z| ¶ R2)

where

R2 = R2(γ) =
3+

�

�γ− 1
�

�−
Æ

9− 2
�

�γ− 1
�

�+
�

�γ− 1
�

�

2

2
�

�γ− 1
�

�

Proof. We let p = 1, q = 0, δ = 0, A= 1, B = −1 in Theorem 2.

Corollary 6 ([see 5]). Let the function f (z) be in the class A1 and suppose that g(z) ∈ C0
1.0(1,1,−1).

If f (z) is majorized by g(z) in U, then

�

�

� f
′
(z)

�

�

� ¶

�

�

�g
′
(z)

�

�

� (|z| ¶ 1

3
)

Proof. We let limit for γ−→ 1 in Corollary 5 or γ−→ 1
2
γ in Corollary 1.

References

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[2] O Altintaş, Ö Özkan, and H Srivastava. Majorization by starlike functions of complex
order. Complex Variables Theory Appl, 46:207–218, 2001.

[3] O. Altintaş and H Srivastava. Some majorization problems associated with p- valently
starlike and convex functions of complex order. East Asian Math. Journal, 17(2):175–
183, 2001.

[4] T MacGregor. The radius of converity for starlike functions of order 1
2
. Amer. Math. Soc,

14:71–76, 1963.

[5] T MacGregor. Majorization by univalent functions. Duke Math.J, 34:95–102, 1967.

[6] M Nasr and M Aouf. Starlike function of complex order. J. Natur. Sci. Math, 25:1–12,
1985.



REFERENCES 24

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[9] M Robertson. On the theory of univalent functions. Ann. of Math., 37(2):1359–1363,
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[10] H Srivastava, O Altıntaş, and S Serenbay. Coffecient bounds for certain subclass of
starlike functions of complex order. Applied Mathematics Letters, 24(8), August 2011.

[11] P Wiatrowski. On the coefficients of some family of holomorphic functions. Nauk. Mat-

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