THE KOMATU INTEGRAL OPERATOR AND STRONGLY CLOSE-TO-CONVEX FUNCTIONS ( C O M M U N IC AT E D B Y C L A U D IO C U E VA S ) M

ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3(2011), Pages 209-219.

THE KOMATU INTEGRAL OPERATOR AND STRONGLY CLOSE-TO-CONVEX FUNCTIONS

( C O M M U N IC AT E D B Y C L A U D IO C U E VA S )

M. K. AOUF

Abstract. In this paper we introduce some new subclasses of strongly close- to-convex functions de…ned by using the Komatu integral operator and study their inclusion relationships with the integral preserving properties.

Theorem[section] [theorem]Lemma [theorem]Proposition [theorem]Corollary Re- mark

1. Introduction LetA₁ denote the class of functions of the form :

f(z) =z+ X1 k=2

akz^k (1.1)

which are analytic in the open unit disc U =fz : jzj <1g. If f(z)and g(z) are analytic inU, we say thatf(z)is subordinate tog(z), writtenf gorf(z) g(z), if there exists a Schwarz functionw(z)in U such thatf(z) =g(w(z)).

A functionf(z)2A1 is said to be starlike of order if it satis…es Re

(zf⁰(z) f(z)

)

> (z2U) (1.2)

for some (0 < 1). We denote by S ( ) the subclass of A₁ consisting of functions which are starlike of order in U. Also a function f(z)2A1 is said to be convex of order if it satis…es

Re (

1 + zf⁰⁰(z) f⁰(z)

)

> (z2U) (1.3)

for some (0 < 1). We denote by C( ) the subclass of A1 consisting of all functions which are convex of order in U.

2000Mathematics Subject Classi…cation. 30C45.

Key words and phrases. Komatu integral operator, strongly close-to-convex.

c 2011 Universiteti i Prishtinës, Prishtinë, Kosovë.

Submitted May 20, 2011. Published July 5, 2011.

209

It follows from(1:2)and(1:3)that

f(z)2C( ),zf⁰(z)2S ( ): (1.4) The classesS ( )and C( )are introduced by Robertson [17] (see also Srivastava and Owa [21]).

Letf(z)2A1 andg(z)2S ( ). Thenf(z)2K( ; )if and only if Re

(zf⁰(z) g(z)

)

> (z2U); (1.5)

where0 <1and0 <1. Such functions are called close-to-convex functions of order and type . The class K( ; ) was introduced by Libera [8] (see also Noor and Alkhorasani [13] and Silverman [19]).

Iff(z)2A₁satis…es

arg(zf⁰(z)

f(z) ) <

2 (z2U) (1.6)

for some (0 <1) and (0< 1), thenf(z)is said to be strongly starlike of order and type inU. We denote this byS ( ; ).

Iff(z)2A₁satis…es

arg(1 + zf⁰⁰(z)

f⁰(z) ) <

2 (z2U) (1.7)

for some (0 < 1) and (0 < 1), then we say that f(z) is strongly convex of order and type in U. We denote this class byC( ; )(see also Liu [10] and Nunokawa [14]). In particular, the classesS ( ;0)andC( ;0)have been extensively studied by Mocanu [12] and Nunokawa [14].

It follows from (1.6) and (1.7) that

f(z)2C( ; ),zf⁰(z)2S ( ; ): (1.8) Also we note thatS (1; ) =S ( )andC(1; ) =C( ).

Recently, Komatu [7] introduced a certain integral operator I_a(a > 0; 0) de…ned by

I_af(z) = a ( )

Z1 0

t^{a 2}(log1

t) ¹f(zt)dt (1.9)

(z2U;a >0; 0;f 2A1):

Thus , iff(z)2A1 is of the form (1.1), it is easily seen from (1.9) that I_af(z) =z+

X1 k=2

a

a+k 1 akz^k (a >0; 0): (1.10) Using the above relation, it is easy verify that

z(I_a⁺¹f(z))⁰ =aI_af(z) (a 1)I_a⁺¹f(z) (a >0; 0): (1.11) We note that :

(i) Fora= 1and =n(nis any integer), the multiplier transformationI₁ⁿf(z) = Iⁿf(z)was studied by Flett [5] and Salagean [18];

(ii) For a = 1 and = n(n 2 N₀ 2 f0;1;2; :::g), the di¤erential operator I₁ⁿf(z) =Dⁿf(z)was studied by Salagean [18];

(iii) Fora= 2and =n(nis any integer), the operatorI₂ⁿf(z) =Lⁿf(z)was studied by Uralegaddi and Somanatha [22];

(iv) For a = 2, the multiplier transformation I₂f(z) = I f(z)was studied by Jung et al. [6].

For a >0 and 0, letK^a( ; ; ; A; B) be the class of functionsf(z) 2A₁ satisfying the condition

arg z(I_af(z))⁰ I_ag(z)

!

< 2 (0 <1; 0< 1;z2U); (1.12) for somef(z)2S^a( ; A; B), where

S^a( ; A; B) = (

g2A1: 1 1

z(I_ag(z))⁰ I_ag(z)

! 1 +Az 1 +Bz

)

(0 <1; 1 B < A 1;z2U): (1.13) We note that K₀^a( ;1; ;1; 1) =K( ; ). We also note that K₀^a(0; ;0;1; 1) is the class of strongly close-to-convex functions of order in the sense of Pommerenke [16]. Also the classS₀^a( ; A; B) =S( ; A; B)was studied by Aouf [1].

In the present paper, using the technique of Cho [3], we give some argument properties of analytic functions belonging toA₁ which contain the basic inclusion relationships among the classes K^a( ; ; ; A; B). The integral preserving prop- erties in connection with the operators I_a de…ned by (1.10) are also considered.

Furthermore, we obtain the previous results given by Bernardi [2] and Libera [9] as special cases.

2. Main Results

In proving our main results, we need the following lemmas.

Lemma 1. [4]. Leth(z) be convex univalent inU withh(0) = 1andRef h(z) + g>0 ( ; 2C). Ifp(z)is analytic inU with p(0) = 1, then

p(z) + zp⁰(z)

p(z) + h(z) (z2U);

implies

p(z) h(z) (z2U):

Lemma 2. [11]. Leth(z)be convex univalent inU andw(z)be analytic inU with Rew(z) 0. Ifp(z)is analytic inU andp(0) =h(0), then

p(z) +w(z)zp⁰(z) h(z) (z2U); implies

p(z) h(z) (z2U):

Lemma 3. [15]. Let p(z) be analytic in U with p(0) = 1 and p(z)6= 0 in U. If there exist two points z₁; z₂2U such that

2 ¹= argp(z1)<argp(z)<argp(z2) =

2 ² (2.1)

for some ₁; ₂( ₁; ₂>0)and for all z(jzj<jz₁j=jz₂j), then we have z₁p⁰(z₁)

p(z₁) = i ¹+ ₂

2 m and z₂p⁰(z₂)

p(z₂) =i ¹+ ₂

2 m; (2.2)

where

m 1 jcj

1 +jcj and c=itan

4( ^{2 1}

1+ 2

): (2.3)

At …rst, with the help of Lemma 1, we obtain the following :

Proposition 4. Let a 1 andh(z) be convex univalent in U with h(0) = 1and Reh(z)>0. If a functionf(z)2A₁ satis…es the condition

1 1

z(I_af(z))⁰ I_af(z)

!

h(z) (0 <1;z2U);

then

1 1

z(I_a⁺¹f(z))⁰ Ia⁺¹f(z)

!

h(z) (0 <1;z2U):

Proof. Let

p(z) = 1 1

z(I_a⁺¹f(z))⁰ Ia⁺¹f(z)

!

(z2U); (2.4)

wherep(z)is analytic function inU withp(0) = 1. By using(1:11), we get a 1 + + (1 )p(z) =a I_af(z)

Ia⁺¹f(z) : (2.5)

Di¤erentiating (2.5) logarithmically with respect to z and multiplying by z, we obtain

p(z) + zp⁰(z)

a 1 + + (1 )p(z) = 1 1

z(I_af(z))⁰ I_af(z)

!

(z2U):

By using Lemma 1, it follows thatp(z) h(z), that is, 1

1

z(I_a⁺¹f(z))⁰ Ia⁺¹f(z)

!

h(z) (z2U):

Takingh(z) = 1 +Az

1 +Bz( 1 B < A 1), in Proposition 1, we have

Corollary 5. The inclusion relation, S^a( ; A; B) S^a₊₁( ; A; B), holds for any a >0 and 0.

Proposition 6. Let h(z)be convex univalent inU withh(0) = 1andReh(z)>0.

If a functionf(z)2A₁ satis…es the condition 1

1

z(I_af(z))⁰ I_af(z)

!

h(z) (0 <1;z2U);

then

1 1

z(I_aL f(z))⁰ I_aL f(z)

!

h(z) (0 <1;z2U); whereL (f)is the integral operator de…ned by

L (f) =L f(z) = + 1 z

Zz 0

t ¹f(t)dt ( 0): (2.6)

Proof. From (2.6), we have

z(I_aL f(z))⁰ = ( + 1)I_af(z) I_aL (f)(z): (2.7) Let

p(z) = 1 1

z(I_aL f(z))⁰ I_aL f(z)

!

(z2U);

wherep(z)is analytic function inU withp(0) = 1. Then, by using (2.7), we have + + (1 )p(z) = ( + 1) I_af(z)

I_aL (f)(z) : (2.8) Di¤erentiating (2.8) logarithmically with respect tozand multiplying byz, we have

p(z) + zp⁰(z)

+ + (1 )p(z) = 1 1

z(I_af(z))⁰ I_af(z)

!

(z2U):

Therefore, by using Lemma 1, we obtain that 1

1

z(I_aL f(z))⁰ I_aL f(z)

!

h(z) (z2U):

Takingh(z) = 1 +Az

1 +Bz( 1 B < A 1), in Proposition 2, we have immediately :

Corollary 7. If f(z)2S^a( ; A; B), thenL (f)2S^a( ; A; B), whereL (f)is the integral operator de…ned by (2.6).

We now derive:

Theorem 8. Let f(z)2A1 and0< 1; 2 1;0 <1. If

2 ¹<arg z(I_af(z))⁰ I_ag(z)

!

< 2 ² for someg(z)2S^a( ; A; B), then

2 ¹<arg z(I_a⁺¹f(z))⁰ Ia⁺¹g(z)

!

< 2 ²; where 1 and 2(0< 1; 2 1) are the solutions of the equations

1= 8>

><

>>

:

1+ ²tan ^{1 ( 1+ 2)(1 jcj) cos2t1}

2(⁽¹ 1+B^)(1+A)+ +a 1)⁽¹⁺jcj)+( 1+ 2)(1 jcj) sin₂t1

for B6= 1;

1 for B= 1;

(2.9) and

2= 8>

><

>>

:

2+ ²tan ^{1 ( 1+ 2)(1 jcj) cos2t1}

2(⁽¹ 1+B^)(1+A)+ +a 1)⁽¹⁺jcj)+( 1+ 2)(1 jcj) sin₂t1

for B6= 1;

2 for B= 1;

(2.10)

wherec is given by (2.3) and t1= 2

sin ¹ (1 )(1 B)

(1 )(1 AB) + ( +a 1)(1 B²) : (2.11) Proof. Let

p(z) = 1 1

z(I_a⁺¹f(z))⁰ Ia⁺¹g(z)

! : Using the identity (1.11) and simplifying, we have

[(1 )p(z) + ]I_a⁺¹g(z) =aI_af(z) (a 1)I_a⁺¹f(z): (2.12) Di¤erentiating (2.12) with respect toz and multiplying byz, we obtain

(1 )zp⁰(z)I_a⁺¹g(z)+[(1 )p(z)+ ]z(I_a⁺¹g(z))⁰ =az(I_af(z))⁰ (a 1)z(I_a⁺¹f(z))⁰ : (2.13) Sinceg(z)2S^a( ; A; B), from Corollary 1, we know thatg(z)2S^a₊₁( ; A; B). Let

q(z) = 1 1

z(I_a⁺¹g(z))⁰ Ia⁺¹g(z)

!

(z2U):

Then, using the identity (1.11) once again, we have (1 )q(z) + +a 1 =a I_ag(z)

Ia⁺¹g(z): (2.14)

From (2.13) and (2.14), we obtain 1

1

z(I_af(z))⁰ I_ag(z)

!

=p(z) + zp⁰(z)

(1 )q(z) + +a 1 ; while, by using the result of Silverman and Silvia [20], we have

q(z) 1 AB

1 B² < (A B)

1 B² (z2U;B 6= 1); (2.15) and

Refq(z)g> 1 A

2 (z2U;B= 1): (2.16)

Then, from (2.15) and (2.16), we obtain

(1 )q(z) + +a 1 = e^{i 2'} ; where

(1 )(1 A)

1 B + +a 1< < ⁽¹ _1+B^)(1+A)+ +a 1; t1< ' < t1 for B6= 1 ;

whent₁ is given by (2.11), and

(1 )(1 A)

2 + +a 1< <1; 1< ' <1 for B= 1:

Here, we note that p(z) is analytic in U with p(0) = 1 and Rep(z) > 0 in U by applying the assumption and Lemma 2 with w(z) = 1

(1 )q(z) + +a 1. Hence p(z)6= 0in U. If there exist two points z₁; z₂ 2U such that the condition

(2.1) is satis…ed, then (by Lemma 3) we obtain (2.2) under the restriction (2.3). At

…rst, for the caseB6= 1, we obtain : arg p(z1) + z1p⁰(z1)

(1 )q(z1) + +a 1

!

= 2 ¹+ arg 1 i ¹+ 2

2 m( e^{i 2} ) ¹ 2 ¹ tan ¹ ( ₁+ ₂)msin ₂(1 ')

2 + ( 1+ 2)mcos₂(1 ')

2 ¹ tan ¹ 8<

:

( ₁+ ₂)(1 jcj) cos₂t₁

2 ⁽¹ _1+B^)(1+A)+ +a 1 (1 +jcj) + ( 1+ 2)(1 jcj) sin₂t1

9=

;

= 2 ¹ ; and

arg p(z₂) + z₂p⁰(z₂)

(1 )q(z2) + +a 1

!

2 ²+ tan ¹ 8<

:

( 1+ 2)(1 jcj) cos ₂t1

2 ⁽¹ _1+B^)(1+A)+ +a 1 (1 +jcj) + ( ₁+ ₂)(1 jcj) sin₂t₁ 9=

;

= 2 ² ;

where we have used the inequality (2.3), and 1; 2andt1are given by (2.9), (2.10) and (2.11), respectively. Similarly, for the caseB= 1, we obtain

arg p(z1) + z1p⁰(z1)

(1 )q(z₁) + +a 1

!

2 ¹ and

arg p(z₂) + z₂p⁰(z₂)

(1 )q(z2) + +a 1

!

2 ² :

These are contradiction to the assumption of Theorem 1. This completes the proof of Theorem 1.

Taking 1= 2= in Theorem 1, then we obtain :

Corollary 9. The inclusion relation,K^a( ; ; ; A; B) K^a₊₁( ; ; ; A; B)holds for any a >0 and 0.

Taking = 0; a= 1and ₁= ₂= in Theorem 1, we obtain : Corollary 10. Let f(z)2A₁. If

arg(zf⁰(z)

g(z) ) <

2 (0 <1;0< 1);

for someg2S( ; A; B), then

arg( f(z)

I₁¹g(z) ) <

2 ;

where (0< 1) is the solution of the equation :

= 8>

><

>>

:

+²tan ^{1 cos2t1}

(⁽¹ _1+B^)(1+A)+ )+ sin₂t1

; for B 6= 1;

for B= 1;

wheret1 is given by (2.11) witha= 1.

Putting = = 0; a = 1; B ! A(A < 1), and g(z) = z in Theorem 1, we obtain

Corollary 11. Let f(z)2A1 and0< 1; 2 1. If 2 ¹<argf⁰(z)<

2 ² ; then

2 ¹<argf(z) z <

2 ² ;

where 1 and 2(0< 1; 2 1) are the solutions of the equations :

1= ₁+2

tan ¹( 1+ 2)(1 jcj) 2(1 +jcj) and

2= ₂+ 2

tan ¹( ₁+ ₂)(1 jcj) 2(1 +jcj) : Next, we prove

Theorem 12. Let f(z)2A1 and0< 1; 2 1;0 <1. If

2 ¹<arg z(I_af(z))⁰ I_ag(z)

!

< 2 ² for someg(z)2S^a( ; A; B), then

2 ¹<arg z(I_aL (f)(z))⁰ I_aL (g)(z)

!

< 2 ² ;

where L (f) is de…ned by (2.6), and 1 and 2(0< 1; 2 1) are the solutions of the equations :

1= 8>

><

>>

:

1+ ²tan ^{1 ( 1+ 2)(1 jcj) cos2t2}

2(⁽¹ 1+B^)(1+A)+ + )⁽¹⁺jcj)+( 1+ 2)(1 jcj) sin₂t2 for B6= 1;

1 for B= 1;

and

2= 8>

><

>>

:

2+ ²tan ^{1 ( 1+ 2)(1 jcj) cos2t2}

2(⁽¹ 1+B^)(1+A)+ + )⁽¹⁺jcj)+( 1+ 2)(1 jcj) sin₂t2

for B6= 1;

2 for B= 1;

wherec is given by (2.3) andt₂ is given by t2= 2

sin ¹ (1 )(A B)

(1 )(1 AB) + ( + )(1 B²) : (2.17)

Proof. Let

p(z) = 1 1

z(I_aL (f)(z))⁰ I_aL (g)(z)

!

(z2U):

Sinceg(z)2S^a( ; A; B), we have from Corollary 2 thatL (g)2S^a( ; A; B). Using (2.7) we obtain

[(1 )p(z) + ]I_aL (g)(z) = ( + 1)I_af(z) I_aL (f)(z):

Then, by a simple calculation, we get

(1 )zp⁰(z) + [(1 )p(z) + ][(1 )q(z) + + ] =

( + 1)z(I_af(z))⁰ I_aL (g)(z) ; where

q(z) = 1 1

z(I_aL (g)(z))⁰ I_aL (g)(z)

! : Hence we have

1 1

z(I_af(z))⁰ I_ag(z)

!

=p(z) + zp⁰(z)

(1 )q(z) + + :

The remaining part of the proof of Theorem 2 is similar to that of Theorem 1 and so we omit it.

Taking 1= 2= in Theorem 2, we have

Corollary 13. Let f(z)2A₁ and0 <1;0< 1. If

arg z(I_af(z))⁰ I_ag(z)

!

< 2 for someg(z)2S^a( ; A; B), then

arg z(I_aL (f)(z))⁰ I_aL (g)(z)

!

< 2 ;

whereL (f)is de…ned by (2.6), and (0< 1)is the solution of the equation:

= 8>

<

>:

+²tan ¹( ^cos2t2

(⁽¹ 1+B^)(1+A)+ + )+ sin₂t2

); for B 6= 1;

for B= 1;

wheret₂ is given by (2.17).

From Corollary 6, we see easily the following corollary.

Corollary 14. f(z) 2 K^a( ; ; ; A; B) =) L (f) 2 K^a( ; ; ; A; B), where L (f) is the integral operator de…ned by (2.6) and is the solution of equation in Corollary 6.

Taking = 0; = 1; A= 1andB = 1in Corollary 7, we obtain :

Corollary 15. Let f(z)2A₁. If Re

(zf⁰(z) g(z)

)

> (0 <1);

then

Re

(z(L (f)(z))⁰ L (g)(z)

)

> (0 <1);

whereL (f)is the integral operator de…ned by (2.6) andg(z)2S ( ) (0 <1).

Remark 1. Taking = = = 0; A = = 1 and B = 1 in Corollary 7, we obtain the classical result obtained by Bernardi [2], which implies the result studied by Libera [8].

2.1. Acknowledgements. The author would like to thank the referee of the paper for his helpful suggestions.

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Department of Mathematics,, Faculty of Science,, Mansoura University,, Mansoura 35516, Egypt.

E-mail address: [email protected]