The order of convexity of some general integral operators

Computers and Mathematics with Applications 62 (2011) 4667–4673

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The order of convexity of some general integral operators Vasile Marius Macarie a,∗ , Daniel Breaz b a

University of Piteşti, Department of Mathematics, Argeş, Romania

b

‘‘1 Decembrie 1918’’, University of Alba Iulia, Department of Mathematics, Alba Iulia, Str. N. Iorga, 510000, No. 11–13, Romania

article

abstract

info

Article history: Received 25 February 2011 Accepted 17 October 2011

The main object of the present paper is to discuss some extensions of certain integral operators and to obtain their order of convexity. Several other closely related results are also considered. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Integral operator Analytic Convexity

1. Introduction Let U = {z ∈ C : |z | < 1} be the unit disk of the complex plane and denote by H (U ) the class of the olomorphic functions in U. Consider A = {f ∈ H (U ) : f (z ) = z + a2 z 2 + a3 z 3 + · · · , z ∈ U } be the class of analytic functions in U and S = {f ∈ A : f is univalent in U }. Consider S ∗ the class of starlike functions in the unit disk, defined by S∗ =





f ∈ A : Re

zf ′ (z )



f (z )

 > 0, z ∈ U .

Definition 1.1. A function f ∈ S is a starlike function of order α , 0 ≤ α < 1 and denote this class by S ∗ (α) if f verifies the inequality

 Re

zf ′ (z )



f (z )

> α,

(z ∈ U ).

Denote with K the class of convex functions in U, defined by

 K =

 f ∈ A : Re

zf ′′ (z ) f ′ (z )

  + 1 > 0, z ∈ U .

Definition 1.2. A function f ∈ S is a convex function of order α , 0 ≤ α < 1 and denote this class by K (α) if f verifies the inequality

 Re

∗

zf ′′ (z ) f ′ (z )

 + 1 > α,

(z ∈ U ).

Corresponding author. E-mail addresses: [email protected] (V.M. Macarie), [email protected] (D. Breaz).

0898-1221/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2011.10.053

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V.M. Macarie, D. Breaz / Computers and Mathematics with Applications 62 (2011) 4667–4673

It is well known that K (α) ⊂ S ∗ (α) ⊂ S. Recently, Frasin and Jahangiri in [1] defined the family B(µ, α), µ ≥ 0, 0 ≤ α < 1 so that it consists of functions f ∈ A satisfying the condition

   µ  ′  z f (z ) − 1 < 1 − α,  f (z )

(z ∈ U ).

(1)

The family B(µ, α) is a comprehensive class of analytic functions which includes various new classes of analytic univalent functions as well as some very well-known ones. For example, B(1, α) ≡ S ∗ (α). In this paper we will obtain the order of convexity of the following general integral operators: G(z ) =

 z n 0

Gn (z ) =

 z n 0

H (z ) =

(2)

(gi (u))βi −1 du,

u

i =1

β  z n  gi (u) i 0

(3)

i=1

1  z n  g i ( u) β 0

F (z ) =

(gi (u))β−1 du,

i =1

u

i=1

un(β−1) du,

(4)

du,

(5)

where the functions g1 (u), g2 (u), . . . , gn (u) are in B(µ, α). In order to prove our main results, we recall the following lemma: Lemma 1.1 ([2] General Schwarz Lemma). Let the function f be regular in the disk UR = {z ∈ C : |z | < R}, with |f (z )| < M for fixed M. If f has one zero with multiplicity order bigger than m for z = 0, then

|f (z )| ≤

M Rm

· |z |m (z ∈ UR ).

The equality can hold only if f (z ) = eiθ ·

M

· zm,

Rm

where θ is constant. 2. Main results Theorem 2.1. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ Mi (Mi ≥ 1, z ∈ U ), for all i = 1, 2, . . . , n then the integral operator G(z ) =

 z n 0

(gi (u))β−1 du

i =1

is in K (δ), where

δ = 1 − |β − 1| · (2 − α)

n 

µ−1

Mi

(6)

i =1

and |β − 1| · (2 − α)

µ−1

< 1, β ∈ C. Proof. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. It follows from (2) that n  gi′ (z ) G′′ (z ) = (β − 1 ) G′ (z ) gi (z ) i=1 n

i=1

Mi

and, hence

  ′′   n    zgi′ (z )   zG (z )    ≤ |β − 1|    G′ (z )   g (z )  i =1

i

 µ     n    ′   gi (z ) µ−1  z g (z ) · ≤ |β − 1|  .  i    g (z ) z 

i =1

i

(7)

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4669

   g (z )  Applying the General Schwarz lemma, we have  i z  ≤ Mi , (z ∈ U ) for all i = 1, 2, . . . , n. Therefore, from (7), we obtain    ′′  µ   n    zG (z )   ′  z µ− 1   g (z ) ·M ,  G′ (z )  ≤ |β − 1|  i  i g (z )

z ∈ U.

(8)

i

i=1

From (1) and (8), we see that

 ′′  n   zG (z )  µ−1   Mi = 1 − δ.  G′ (z )  ≤ |β − 1| · (2 − α) i=1

This completes the proof.



For M1 = M2 = · · · = Mn = M, we have Corollary 2.1. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) then the integral operator G(z ) =

 z n (gi (u))β−1 du 0

i=1

is in K (δ), where

δ = 1 − |β − 1|n(2 − α)M µ−1 and |β − 1|n(2 − α)M

µ−1

(9)

< 1, β ∈ C.

Letting δ = 0 in Corollary 2.1 we have Corollary 2.2. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) then the integral operator defined in (2) is convex function in U, where 1

|β − 1| =

n(2 − α)M µ−1

,

β ∈ C.

Letting µ = 1 in Corollary 2.1, we have Corollary 2.3. Let gi (z ) be in the class S ∗ (α), 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) then the integral operator defined in (2) is in K (δ), where

δ = 1 − |β − 1|n(2 − α)

(10)

and |β − 1|n(2 − α) < 1, β ∈ C. Letting n = 1 and α = δ = 0 in Corollary 2.3, we have Corollary 2.4. Let g (z ) be a starlike function in U. If |g (z )| ≤ M (M ≥ 1, z ∈ U ) then the integral operator 1 2

convex in U, where |β − 1| =

, β ∈ C.

z 0

g (u)β−1 du is

Theorem 2.2. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ Mi (Mi ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator

 z n

Gn (z ) =

0

(gi (u))βi −1 du

i =1

is in K (δ), where

δ = 1 − (2 − α)

n 

µ−1

|βi − 1|Mi

i =1

and (2 − α)

n

µ−1

i =1

|βi − 1|Mi

< 1, βi ∈ C for all i = 1, 2, . . . , n.

Proof. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1. It follows from (3) that G′′n (z ) G′n (z )

=

n  (βi − 1)g ′ (z ) i

i=1

gi (z )

(11)

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V.M. Macarie, D. Breaz / Computers and Mathematics with Applications 62 (2011) 4667–4673

and, hence

 ′′    ′  n  zGn (z )   zg (z )   ≤ |βi − 1|  i   G′ (z )  gi (z ) n i=1  µ      n   ′   gi (z ) µ−1  z   ≤ |βi − 1| · gi (z ) .  ·   g (z ) z

(12)

i

i =1

   gi (z )   ≤ Mi , (z ∈ U ) for all i = 1, 2, . . . , n. Therefore, from (12), we obtain z

Applying the General Schwarz lemma, we have 

 ′′    µ   n   zGn (z )   ′ z  · M µ−1 , ≤   |βi − 1| · gi (z )  i  G′ (z )  g ( z ) i n i=1

(z ∈ U ).

(13)

From (1) and (13), we see that

 ′′  n   zGn (z )  µ−1   |βi − 1| · Mi = 1 − δ.  G′ (z )  ≤ (2 − α) n i=1 This completes the proof.



For M1 = M2 = · · · = Mn = M, we have Corollary 2.5. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator Gn (z ) =

 z n 0

(gi (u))βi −1 du

i=1

is in K (δ), where

 δ =1−

n 

 |βi − 1| (2 − α)M µ−1

(14)

i =1

 |βi − 1| (2 − α)M µ−1 < 1, βi ∈ C for all i = 1, 2, . . . , n. Letting δ = 0 in Corollary 2.5, we have

and

n

i=1

Corollary 2.6. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator defined in (3) is a convex function in U, where n 

|βi − 1| =

i =1

1

(2 − α)M µ−1

,

βi ∈ C for all i = 1, 2, . . . , n.

Letting µ = 1 in Corollary 2.5, we have Corollary 2.7. Let gi (z ) be in the class S ∗ (α), 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator defined in (3) is in K (δ), where

δ =1−

n 

|βi − 1| · (2 − α)

(15)

i =1

and

n

i=1

|βi − 1| · (2 − α) < 1, βi ∈ C for all i = 1, 2, . . . , n.

Theorem 2.3. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ Mi (Mi ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator H (z ) =

1  z n  g i ( u) β 0

u

i=1

un(β−1) du

is in K (δ), where

 δ =1− 1 and |β|

n 1 

|β|

µ−1

(2 − α)Mi





+ 1 + n|β − 1|

i=1

 n  µ−1 + 1 + n|β − 1| < 1, β ∈ C \ {0}. i=1 (2 − α)Mi

(16)

V.M. Macarie, D. Breaz / Computers and Mathematics with Applications 62 (2011) 4667–4673

4671

Proof. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1, for all i = 1, 2, . . . , n. It follows from (4) that H ′′ (z ) H ′ (z )

1

=

g1′ (z ) · z − g1 (z )

·

β

zg1 (z )

+ ··· +

1

β

·

gn′ (z ) · z − gn (z ) zgn (z )

+

n(β − 1) z

and zH ′′ (z ) H ′ (z )

=

1

β

·

n  ′  g (z ) · z i

gi (z )

i=1

 − 1 + n(β − 1).

So that

 ′′    n    zH (z )   zgi′ (z )   ≤ 1 ·   + 1 + n|β − 1|  H ′ (z )  |β| i=1  gi (z )    µ      n   gi (z ) µ−1  z 1   ′   · g (z ) ≤ (17)  + 1 + n|β − 1|.  ·   |β| i=1  i gi (z ) z    g (z )  Applying the General Schwarz lemma, we have  i z  ≤ Mi , (z ∈ U ) for all i = 1, 2, . . . , n. Therefore, from (17), we obtain  ′′    µ  n    zH (z )   ′  z ≤ 1 ·  g (z )  · M µ−1 + 1 + n|β − 1|,  H ′ (z )  |β|  i  i gi (z ) i=1

(z ∈ U ).

(18)

From (1) and (18), we see that

 ′′  n     zH (z )  µ−1  ≤ 1 + 1 + n|β − 1| = 1 − δ. ( 2 − α) M i  H ′ (z )  |β| i =1 This completes the proof.



For M1 = M2 = · · · = Mn = M, we have Corollary 2.8. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator H (z ) =

1  z n  g i ( u) β 0

i =1

u

un(β−1) du

is in K (δ), where

δ =1−



n 

|β|

 (2 − α)M µ−1 + 1 + n|β − 1|

 (19)

n and |β| (2 − α)M µ−1 + 1 + n|β − 1| < 1, β ∈ C \ {0}.





Letting δ = 0 in Corollary 2.8 we have Corollary 2.9. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator defined in (4) is a convex function in U, where n 

|β|

 (2 − α)M µ−1 + 1 + n|β − 1| = 1,

β ∈ C \ {0}.

Letting µ = 1 in Corollary 2.8, we have Corollary 2.10. Let gi (z ) be in the class S ∗ (α), 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator defined in (4) is in K (δ), where

δ =1− 



n

|β|

 (3 − α) + n|β − 1| 

n and |β| (3 − α) + n|β − 1| < 1 for all β ∈ C \ {0}.

(20)

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Letting n = 1 and α = δ = 0 in Corollary 2.10, we have Corollary 2.11. Let g (z ) be a starlike function in U. If |g (z )| ≤ M (M

 z  g (u)  β1 0

u

≥ 1, z ∈ U ) then the integral operator

3 uβ−1 du is convex in U, where |β| + |β − 1| = 1, β ∈ C \ {0}.

Theorem 2.4. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ Mi (Mi ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator F (z ) =

β  z n  gi (u) i 0

du

u

i=1

is in K (δ), where

δ =1−

n 

µ−1

|βi | · [(2 − α)Mi

+ 1]

(21)

i =1

and

n

i=1

µ−1

|βi | · [(2 − α)Mi

+ 1] < 1, βi ∈ C for all i = 1, 2, . . . , n..

Proof. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1. It follows from (5) that zF ′′ (z ) F ′ (z )

=

n 

βi



zgi′ (z )

 −1 .

gi (z )

i =1

(22)

It follows from (22)

 ′′    ′   n  zF (z )   zgi (z )   ≤   |β | + 1 i  F ′ (z )   g (z )  i

i=1

   µ     n   ′   gi (z ) µ−1  z   ≤ |βi | gi (z ) +1 .  ·   g (z ) z

(23)

i

i=1

   gi (z )   ≤ Mi , (z ∈ U ) for all i = 1, 2, . . . , n. Therefore, from (23), we obtain z

Applying the General Schwarz lemma, we have 

 ′′      µ  n  zF (z )   ′  z   ≤  · M µ−1 + 1 , g |β | ( z ) i  F ′ (z )   i  i g (z )

(z ∈ U ).

(24)

i

i =1

From (1) and (24), we see that

 ′′   n    zF (z )  µ−1  ≤ |β | ( 2 − α) M + 1 = 1 − δ. i i  F ′ (z )  i =1

This completes the proof.



For M1 = M2 = · · · = Mn = M, we have Corollary 2.12. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator F (z ) =

β  z n  gi (u) i 0

i=1

u

du

is in K (δ), where

δ =1−

n 

|βi | · [(2 − α)M µ−1 + 1]

i =1

and

n

i=1

|βi | · [(2 − α)M µ−1 + 1] < 1, βi ∈ C for all i = 1, 2, . . . , n.

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4673

Letting δ = 0 in Corollary 2.12, we have Corollary 2.13. Let gi (z ) be in the class B(µ, α), µ ≥ 1, 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator defined in (5) is a convex function in U, where n 

|βi | =

i =1

1

(2 − α)M µ−1 + 1

,

βi ∈ C for all i = 1, 2, . . . , n.

Letting µ = 1 in Corollary 2.12, we have Corollary 2.14. Let gi (z ) ∈ A be in the class S ∗ (α), 0 ≤ α < 1 for all i = 1, 2, . . . , n. If |gi (z )| ≤ M (M ≥ 1, z ∈ U ) for all i = 1, 2, . . . , n, then the integral operator defined in (5) is in K (δ), where

δ =1−

n 

|βi |(3 − α)

(26)

i=1

and

n

i=1

|βi |(3 − α) < 1, βi ∈ C for all i = 1, 2, . . . , n.

Letting n = 1 and α = δ = 0 in Corollary 2.14, we have Corollary 2.15. Let g (z ) ∈ A be a starlike function in U. If |g (z )| ≤ M (M ≥ 1, z ∈ U ) then the integral operator

 z  g (u) β 0

u

du is convex in U, where |β| =

1 3

, β ∈ C.

References [1] B.A. Frasin, J. Jahangiri, A new and comprehensive class of analytic functions, An. Univ. Oradea Fasc. Mat. XV (2008) 59–62. [2] Z. Nehari, Conformal Mapping, Dover, New York, NY, USA, 1975.