FUNCTIONS OF BOUNDED RADIUS ROTATION OF ANALYTIC FUNCTIONS by IASET Journals

International Journal of Applied Mathematics & Statistical Sciences (IJAMSS)

ISSN (P): 2319–3972; ISSN (E): 2319–3980

Vol. 11, Issue 2, Jul–Dec 2022; 43–50

FUNCTIONS OF BOUNDED RADIUS ROTATION OF ANALYTIC FUNCTIONS

S.Prema1 & Bhuvaneswari Raja2

1Assistant Professor, Department of Mathematics, Rajalakshmi Institute of Technology, Chennai-600124, India

ABSTRACT

The classes )( α kV , )( α kR , )( 1 α kV and )( 1 α kR of analytic function, We establish a relation between the functions of bounded boundary and bounded radius rotations

KEYWORDS: Analytic, Analytic Functions with Bounded Radius and Boundary Rotations, Starlike, Convex, Subordination

MATHEMATICS SUBJECT CLASSIFICATION: 30C45

Article History

Received: 16 Aug 2022 | Revised: 17 Aug 2022 | Accepted: 24 Aug 2022

1. INTRODUCTION

Let A be the class of functions f of the form

Which are analytic in the unit disc E = {z: |z| < 1}.

The definition of subordination is ∈ f A is subordinate to g∈A, written as gf p ,there exists Schwarz function w(z) with w(0) = 0 and |w(z)| < 1 ( ∈ z E) such that f(z) = g(w(z)), In particular, when g is univalent, then the above subordination is equivalent to f(0)’ = g(0) and f(E) ⊆ g(E), for any two analytic functions

then convolution

We denote by )10(),(),( * <≤ α α α cs the classes of starlike and convex functions of order α,

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∑ ∞ = += 2 )( n n n zazzf (1.1)

).()(,)( 00 Ezzbzgzazf n n n n n n ∈==∑∑ ∞ = ∞ = (1.2)

)Ez())(*( 0 ∈= ∑ ∞ =n n nn zbazgf (1.3)

2Assistant Professor, Department of Mathematics, Aalim Muhammed Salegh College of Engineering, Chennai, India.

respectively defined by

For 0 = α , we have the well know classes of starlike and convex univalent functions denoted * s and c, respectively.

Let )( α kP be the class of functions P(z) analytic in the unit disc E satisfying the properties P(0)=1 and

Where ,2,≥=krez iθ and 10 < ≤ α , using Herglotz – Stieltijes formula

Where )( α P is the class of functions with real part greater then α and ),( α PP i ∈ for i = 1, 2 we define the following classes

44 S.Prem1 & Bhuvaneswari Raja

{} EzszfzAfc Ez zf zfz Afs ∈∈ ′ ∈=       ∈> ′ ∈= ),()(;)( , )( )( Re:)( * * α α α α (1.4)

, 1 )( Re 2 0 π θ α α π kd zP ≤ ∫ (1.5)

EzzP k zP k zP ∈             += ),( 2 1 4 )( 2 1 4 )(21 (1.6)

[ ] [] []0t,10),( )()()(1())()(( )()(( :)( 0>t,10),( ))()()(1())()(( )()( :)( 2 2 >           <≤∈ ′ ′ ′ −+− ′′ ′′ ′ ′ + ′′ ∈=       <≤∈ −−−+− ′ ′ ′ + ′′ ∈= λ α λ λ λ α λ α λ λ λ α k k k k P zfzfzfzfz zfzzfz andAffV P zfzfzfzfz zfzzfz andAffR (1.7) [] .10 ),( ))()()(1( )()(()())()((:)()()()2()( 2 23 1 <≤≤               ∈ −−+−+ ′ ′ −+− ′′ ′′ ′ + ′′ +++ ∈= γ λ α γ λ γ λ γλ λ γ γλ γλ α k k P zfzf zfzfzzfzfz andAffRzfzzfzzfz (1.8) [] [] .10 ),( )()()(1( )()(()())()((:)()()()2()( 2 23 1 <≤≤               ∈ ′ −−+−+ ′ ′ −+− ′′ ′′ ′ ′ + ′′ +++ ∈= γ λ α γ λ γ λ γλ λ γ γλ γλ α k k P zfzf zfzfzzfzfz andAffVzfzzfzzfz (1.9) We note that )()()()( 11 λ λ λ λ kkkkRfzVfRfzVf ∈ ′ ⇔ ∈ ∈ ′ ⇔ ∈ (1.10) Impact Factor (JCC): 6.6810 NAAS Rating 3.45

For 0 = = = γ λ t we obtain the well known classes Rk, Rk1, Vk, Vk1 of analytic functions with bounded radius and bounded boundary rotations, respectively. These classes are studied by Noor [3] in more details, also it can easily

result is sharp. In this paper, we prove the result of Goel [6] for the

by using convolution and Subordination techniques.

2. PRELIMINARY RESULTS

We need the following results to obtain our results.

Functions of Bounded Radius Rotation of Analytic Functions 45 www.iaset.us editor@iaset.us

seen )( )( * 2 α α S Rk = and )( )( 2 α α c V = . Goel [6]

)( α c f ∈ implies that )( * α s f ∈ and )( * α S f ∈ Where      = ≠ = = + 2 1 , 2ln 2 1 2 1 , 2 4 ) 2 1(4 )( 12 α α α α β β α α and

)( α kV , )( α kR , )( 1 α kV and

1 α kR

proved that

the

classes

)(

Lemma 2.1: Let 2 1 2 1 , iv v v iu u u + = + = and),(vuΨ be a complex valued function satisfying the conditions (i)),(vuΨ is continuous in a domain 2C D ⊂ (ii)(1,0) D ∈ and Re)0,1( > ψ when ever 0 ) , ( 1 2 ≤ v iu when ever () 2 2 1 12 1 2 1 and ), ( u v D v iu +       −≤ ∈ If ....... 1 )( 1 + + = zc zh is a function analytic in E. Such that D z zh ∈ ′ )) ( ), (( and 0 )) ( Re),(( > ′ z zhψ for ,E z ∈ then Re h(z)>0 in E.

Theorem 3.1 Let ), (α kV f ∈ then ), (α kR f ∈ where ). 1( 8 8 8 ) 4 8 ) 1( 4 4 4 ) 1( ) 2 ) 1( 4 1 2 2 22 2 2 λ λ λ λ α λ α λ α λ α α λ αλ λ α β + + + + + + + + + + + + + = Proof: Let β β λ λ λ + = + ′ + ′′ )() 1( )) ( )( )( 1( )) ( )(( )) ( )( ( 2 z P z f zf z f zfz z z fz (3.1)

3. MAIN RESULTS

46 S.Prem1 & Bhuvaneswari

= β β +                   +− )( 2 1 4 )( 2 1 4 )1( 21 zP k zP k (3.2) P(z) is analytic in E with P(0)=1 then [ ) [ ] [][] [] β β λ λ λ λ β λ β β λ λ λ +−−+− ′ ′       ′ +−+ ′ −+ ′ + ′′ + +−= −−−+− ′ ′ ′ ′ + ′′ )()1)(1()()( )()1( )()1()()( )()1( )()()1()()( )()( 2 2 zpzfzfz zfzz zpzfztzfz zp zfzfzfzfz zfzzfz (3.3) [ ) [ ] [][] [] [][]                             ++ ′ +         + + =       +−−+− ′ ′ ′ −−+ ′ + ′′ +−+− =         −−−+− ′ ′ ′ ′ + ′′ β β β α β α α β β β λ λ β λ λ λ α β β α α λ λ λ α 1 )()(( )1()()( 1 )( 1 1 1 )()1()1()()( )1()()1()()1.()1()()( 1 )()()1()()( )()( )1( 1 2 2 zpQN zpML zp zpzfzfz zpzzfzzfz zp zfzfzfzfz zfzzfz (3.4) Where )( 2 zfzL ′′ = λ zzfzM)1()( λ λ + ′ = [ ] )()( tzfzfzN ′ ′ = λ )1( λ = Q Since )( α k Vf ∈ , it implies that () () Ezp zpQN zpML zp k ∈∈                     ++ ′ +         + + = , 1( )( )( 1 1 )( 1 1 1 β β β α β α α β (3.5) We define a ba z z b b z z b z + −+ + −+ = 1 , )1(1)1(1 1 )( α ϕ (3.6) with β = 1 1 a , β β = 1 b by using (3.1) with convolution,

Impact Factor (JCC): 6.6810 NAAS Rating 3.45

Raja

see [5], we have, that

and

we have

We now from functional),(vuψ by choosing )( ),( z zp v z p u i i

= = in (3.8) and note that the first two conditions of Lemma 2.1 are likely satisfied, we check the condition as follows

Functions of Bounded Radius Rotation of Analytic Functions 47 www.iaset.us editor@iaset.us                        + = )( * )( 2 1 4 )( * )( 2 1 4 )( * )( 2 , 1 , , z p z z k z p z z k zp z z ba ba ba ϕ ϕ ϕ Implies         + + ′ + +       +         + + ′ + +       + = + + ′ + + b z p Q N z p M L a z p k b z p Q N z p M L a z p k b zp R Q N z p M L a zp )( )( ( )( ) ( )( 2 1 4 )( )( ( )( ) ( )( 2 1 4 ) )()( ( )( ) ( )( 2 2 2 1 1 1 (3.7)

(3.5)

(3.7)

1,2 i )( )( ( )( ) ( )( ) 1( ) 1 = ∈         + + ′ + + + p b z p Q N z p M L a z p i i i α β α α β (3.8)

Thus from

[] [][] () () () () () () () )( )( )(( 1 2 1 )( )( ) 1( ) )( )( )( 1 1 )( )( ) 2 1 ) ( 1 1 )( )( ) 1( )( )( 1 )( )( )( ) ( 1 1 )( )( ) 1( )( )( ( 1 )( )( )( Re ) ( 1 1 ), ( Re 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 2 1 2 12 z fz zf zf u z fz zf u z fz zf zf u z fz zf u zf zf zf z fz iu zvzf pz fz zf zf tz f z fz iu zvzf pz fz v iu i i ′′ +                 +         ′′ + ′ + =                 ′′ +                 +         ′′ + ′ + ≤               ′         + ′ + ′′         +              ′ + ′ ′′ + =                             ′ ′ ′ + ′′ ′′         + ′ + ′ + = λ β β β β λ α β λ β β β β λ α β α λ λ β β β β λ α β α λ λ β β λ α β α ψ (3.9) 2 , 2 2 2 > + = C c Bu A

′

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NAAS Rating 3.45

48 S.Prem1 & Bhuvaneswari Raja Where [ ] ])()1()]()([()1( 1 ])()([])()]()([()1()(2[ )1( 1 )1(])()([])()]()([()(2 )1( 2 2 2 2 22 22 2 ′′′ ++−−                 += ′′ + ′ ′′′′ +−−− = ′′ + ′ ′′′ +−− = zfzzfzfuC zfzzfzfzzfzfB zfzzfzfzzfzfA λ λ α β β λ β λ β α β β β λ λ β α β β β (3.10) The right hand side of (3.9) is negative if 0 ≤ A we have ).1(888 )1(48)1(444)1( )2)1(2( 4 1 )( 2 22222 λ λ λ λ α λ α λ α λ α α λ αλ λ α α β β +++−+ +−++−+++ +++−== Theorem 3.2 Let ),( 1 α k Vf ∈ then ),( 1 α k Rf ∈ where ).1)(1( 12)1)(1(8)1)(1(416 )1)(1(464)1( 2)1)(1(2( 4 1 22 2 222222 λ γ γλ γ λ γ λ α λγ α λ γ α λ γ α λγ α γ λ αγλ γ λ α β +++ −+++++−+ ++−++++ ++++−= t

vol.81, pp.79-81, 1975.

10. S.S.Miller and P.T.Mocanu, “differential subordinations :theory and application ,vol.225of monographs and text bookes in pure and applied mathematicals,” marcel dekker ,new york ,NY,USA,2000

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13. T.H.MacGregor, “A subordination for convex function of order a,” journal of the London mathematical society second series, vol.9, pp.530-536, 1975.

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