For˛Dˇ the classS S

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Vol. 19 (2018), No. 1, pp. 431–437 DOI: 10.18514/MMN.2018.1945

ON STRONGLY STARLIKE FUNCTIONS OF ORDER .˛; ˇ/

MAMORU NUNOKAWA AND JANUSZ SOK ´OŁ Received 04 April, 2016

Abstract. We consider the class S S.˛; ˇ/of analytic functions which satisfy the condition ˇ=2 <arg˚

´f⁰.´/=f .´/ < ˛=2for all´in the unit discEon the complex plane, where 0˛ < 1and 0ˇ < 1. For˛Dˇ the classS S.˛; ˇ/is equal to the well known class S S.ˇ/of strongly starlike functions of orderˇ. In this work we derive a sufficient condition for analytic function to be in the classS S.˛; ˇ/strongly starlike functions of order.˛; ˇ/.

2010Mathematics Subject Classification: 30C45

Keywords: convex functions, starlike functions, starlike of order˛, convex of order˛, strongly starlike functions, subordination

1. INTRODUCTION

LetAdenote the class of analytic functionsf in the unit discED f´W j´j< 1gon the complex planeCwith the normalizationf .0/D0,f⁰.0/D1. A functionf 2A is said to be starlike of orderıif

Re

´f⁰.´/

f .´/

> ı .´2E/; (1.1)

for some0ı < 1, Robertson [8]. We denote byS.ı/the class of functions starlike of orderı. We say that a functionf 2Ais strongly starlike of orderˇif and only if

ˇ ˇ ˇ ˇ

arg

´f⁰.´/

f .´/

ˇ ˇ ˇ ˇ

<

2ˇ .´2E/;

for someˇ .0 < ˇ1/:LetS S.ˇ/denote the class of strongly starlike functions of orderˇ. The classS S.ˇ/was introduced independently by Stankiewicz [9,10] and by Brannan and Kirvan [1]. In [11] Takahashi and Nunokawa defined the following subclass ofA:

S S.˛; ˇ/D

f 2AW ˇ

2 <arg´f⁰.´/

f .´/ < ˛ 2 ; ´2E

;

for some0 < ˛1;and for some0 < ˇ1. We recall here the fact, that in [2] and in [3] a similar class was studied, see also [4, p.141]. Note thatS S.minf˛; ˇg/

c 2018 Miskolc University Press

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S S.˛; ˇ/S S.maxf˛; ˇg/. Of course for˛Dˇ the classS S.˛; ˇ/becomes the classS S.ˇ/. It is easily seen thatS S.˛; ˇ/S.0/.

2. PRELIMINARY RESULTS

Lemma 1([5]). Letw.´/DaCwn´ⁿCwnC1´ⁿ^C¹C be analytic in Ewith w.´/6aandn1. If´0Dr0eⁱ,0 < r0< 1and

jw.´0/j D max

j´jr₀jw.´/j; then it follows that

´0w⁰.´0/ w.´0/ Dm;

where

mn jw.´0/ aj²

jw.´0/j² jaj² njw.´0/j jaj jw.´0/j C jaj:

Theorem 1. Letpbe analytic inEwithp.0/D1andp.´/¤0. If there exist two points´12Eand´22Esuch thatj´1j D j´2j Dr < 1and for´2ErD f´W j´j< rg

ˇ

2 Dargp.´1/ <argp.´/ <argp.´2/D ˛

2 ; (2.1)

with some0 < ˛2,0 < ˇ2, then we have

´1p⁰.´1/

p.´1/ D i m

˛Cˇ 2

1Cs²

2s (2.2)

and

´2p⁰.´2/ p.´2/ Di m

˛Cˇ 2

1Ct²

2t ; (2.3)

where

m 1 jaj

1C jaj; aDitan 4

˛ ˇ

˛Cˇ

and where

sD jp.´1/j^2=.˛^C^{ˇ /}; t D jp.´2/j^2=.˛^C^{ˇ /}:

Proof. The assumption (2.1) says that the domainp.Er/lies in a sector between two rays argfwg D ˇ=2and argfwg D ˛=2and it contacts with the rays atp.´1/ and atp.´₂/. The idea of this proof is that we transform this sector into the unit disc and then we will use the Lemma1. We restrict our considerations to proving (2.3), the proof of (2.2) runs analogously as that of (2.3). The function

q.´/DAfp.´/g^B; .´2Er/; (2.4) where

ADexp

i.˛ ˇ/

2.˛Cˇ/

; B D 2

˛Cˇ;

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mapsEronto the setq.Er/on the right half-planeRef!g> 0. The boundary@q.Er/ is tangent to the imaginary axis atq.´1/and at q.´2/because@p.Er/is tangent to the sector ˇ=2 <argw < ˛=2atp.´1/and atp.´2/. Moreover,q.´1/lies on the negative imaginary axis, whileq.´2/lies on the positive imaginary axis,

argfq.´₁/g Darg n

Afp.´₁/g^Bo

D .˛ ˇ/

2.˛Cˇ/

ˇ 2

2

˛Cˇ D

2 (2.5)

and

argfq.´2/g Dargn

Afp.´2/g^Bo

D .˛ ˇ/

2.˛Cˇ/C˛ 2

2

˛Cˇ D

2; (2.6) with

jargfq.´/gj D ˇ ˇ ˇarg

n

Afp.´/g^Boˇ ˇ ˇ<

2 .´2Er/:

This shows that

Refq.´/g DRen

Afp.´/g^Bo

> 0 .´2Er/:

Therefore, the function

.´/D 1 q.´/

1Cq.´/ .´2Er/ (2.7)

maps theErinto the unitEdisc and satisfies

jmax´jrj.´/j D j.´1/j D j.´2/j D1: (2.8) By logarithmic differentiation of (2.7), we have

´⁰.´/

.´/ D´p⁰.´/

p.´/

2ABp^B.´/

1 A²p^2B.´/: For the case argfp.´2/g D˛=2, from (2.6), we can put

Ap^B.´2/Di t; 0 < t:

Applying Lemma1and (2.8), we have

´2⁰.´2/

.´2/ D ´2p⁰.´2/ p.´2/

2ABp^B.´2/ 1 A²p^2B.´2/

D ´2p⁰.´2/ p.´2/

4i t

˛Cˇ 1

1Ct² (2.9)

Dm where

mn j.´2/ .0/j²

j.´2/j² j.0/j² 1 j.0/j

1C j.0/jD 1 jaj 1C jaj and

aDitan 4

˛ ˇ

˛Cˇ

:

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From (2.9), we obtain (2.3) and for the case argfp.´1/g D ˛=2, applying the same method as the above and from (2.5) and (2.8), we obtain (2.2). It completes the proof

of Theorem1.

The above result is a stronger form of Theorem 1 in [7]. For˛ DˇTheorem 1 becomes Nunokawa’s lemma [6].

3. MAIN THEOREM

Our main result is contained in:

Theorem 2. Letf .´/D´CP₁

nD2an´ⁿbe analytic inE. Assume that0 < ˛1, 0 < ˇ1and

ˇ

2 B.˛; ˇ/ <arg

1C´f⁰⁰.´/

f⁰.´/

< ˛

2 CA.˛; ˇ/ ´2E; (3.1) whereA.1; 1/D0,B.1; 1/D0, while for˛Cˇ < 2

B.˛; ˇ/Dtan ¹ m.˛Cˇ/g.t0/cos.ˇ=2/

2Cm.˛Cˇ/g.t0/sin.ˇ=2/; A.˛; ˇ/Dtan ¹ m.˛Cˇ/g.t0/cos. ˛=2/

2Cm.˛Cˇ/g.t0/sin. ˛=2/; and where,

m 1 jaj

1C jaj; aDitan

˛ ˇ

˛Cˇ

;

g.t /D 1Ct²

2t^.2^C^˛^C^{ˇ /=2}; t0D s

2C˛Cˇ 2 ˛ ˇ: Then we have

ˇ 2 <arg

´f⁰.´/

f .´/

< ˛

2 ´2E; (3.2)

Proof. Let us define the function

p.´/D ´f⁰.´/

f .´/ ; p.0/D1;

then it follows that

p.´/C´p⁰.´/

p.´/ D1C´f⁰⁰.´/

f⁰.´/ : If there exists two points´1; ´22E,j´1j D j´2j Drsuch that

ˇ

2 Dargp.´1/ <argp.´/ <argp.´2/D ˛ 2 ;

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with some0 < ˛1,0 < ˇ1, then for all´2Er, then from Theorem1

´1p⁰.´1/

p.´1/ D i m

˛Cˇ 2

1Cs² 2s and

´2p⁰.´2/ p.´2/ Di m

˛Cˇ 2

1Ct² 2t ; where

m 1 jaj

1C jaj; aDitan 4

˛ ˇ

˛Cˇ

and where

sD jp.´1/j^2=.˛^C^{ˇ /}; t D jp.´2/j^2=.˛^C^{ˇ /}: For the case argfp.´2/g D˛=2, from Theorem1we have

arg

1C´2f⁰⁰.´2/ f⁰.´2/

Darg

p.´2/C´2p⁰.´2/ p.´2/

Dargfp.´2/g Carg

1C´2p⁰.´2/ p.´2/

1 p.´2/

D˛ 2 Carg

1Ci m

˛Cˇ 2

1Ct² 2t

1 t^.˛^C^{ˇ /=2}e^{i ˛=2}

D˛ 2 Carg

1Cm

˛Cˇ 2

1Ct² 2t^.2^C^˛^C^{ˇ /=2}

e^{i .1 ˛/=2}

:

Hence, if ˛D1 then argf1C´2f⁰⁰.´2/=f⁰.´2/g D˛=2, which contradicts (3.1) because in this caseAD0. For0 < ˛ < 1, it easily confirm that

arg

1Cm

˛Cˇ 2

1Ct² 2t^.2^C^˛^C^{ˇ /=2}

e^{i .1 ˛/=2}

:

takes its minimum value when the function g.t /D 1Ct²

2t^.2^C^˛^C^{ˇ /=2}; t > 0;

takes its minimum value att0Dp

.2C˛Cˇ/=.2 ˛ ˇ/and so, we have arg

1C´2f⁰⁰.´2/ f⁰.´2/

˛

2 Ctan ¹ m.˛Cˇ/g.t0/sin..1 ˛/=2/

2Cm.˛Cˇ/g.t0/cos..1 ˛/=2/

This contradicts (3.1) and for the case argfp.´1/g Dˇ=2, applying the same method as the above we obtain forˇD1a contradiction with (3.1), while for0 < ˇ < 1we

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have

arg

1C´1f⁰⁰.´1/ f⁰.´1/

ˇ

2 tan ¹ m.˛Cˇ/g.t0/sin..1 ˇ/=2/

2Cm.˛Cˇ/g.t0/cos..1 ˇ/=2/:

This also contradicts (3.1) and therefore, it completes the proof.

If˛Dˇin the above theorem then we get the following corollary.

Corollary 1. Assume that

4 tan ¹

p2 2p⁴

27Cp 2<arg

1C´f⁰⁰.´/

f⁰.´/

<

4Ctan ¹

p2 2p⁴

27Cp

2 ´2E:

Then we have

4 <arg

´f⁰.´/

f .´/

<

4 ´2E;

this means thatf is strongly starlike of order1=2.

Note that

tan ¹

p2 2p⁴

27Cp

2 D0:23 : : : : REFERENCES

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Authors’ addresses

Mamoru Nunokawa

University of Gunma, Hoshikuki-cho 798-8, Chuou-Ward, Chiba, 260-0808, Japan E-mail address:mamoru nuno@doctor.nifty.jp

Janusz Sok´oł

University of Rzesz´ow, Faculty of Mathematics and Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzesz´ow, Poland

E-mail address:jsokol@ur.edu.pl