CRITICAL INVESTIGATION OF VARIOUSSUBCLASSES OF … · 2019. 5. 13. · CRITICAL INVESTIGATION OF...

Innat. J. Math. Mth. Sci.Vol. 4 No. (1981) 89- 99

89

A CRITICAL INVESTIGATIONOF VARIOUS SUBCLASSES OF FUNCTIONS-

WHOSE REAL PART IS BOUNDED

S.K. BAJPAI

Universidade de BrasiliaDepartamento de Matematica70.910- Brasilia- DF

Brasil

(Received October 22, 1979)

ABSTRACT. In this paper we investigate various subclasses of univalent analytic

functions. We find that many of the subclasses introduced in the recent years are

no more new and infact coincide with the class due to Jakubowski. We further

study the generalised Jakubowski class of univalent functions and obtain a

representational formula and use it in deriving the coefficient relations for

this class.

KEY WORDS AND PHRASES. Univent Stke, onvex, Sp like, Boanded fcurcio,Fanction positive rl pa.

1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. P OA92, Second OA94.

i. INTRODUCTION.,

The Class S (,v).m,M

Let M,m be arbitrary fixed real numbers satisfying the condition

90 S.K. BAJPAI

D*(re,M) E Dlt D2 andI

Jakubowski [7] introduced the class S*,M(’)m of regular functions f(z)z + akzk defined in E {z Iz[ < I} and satisfying the cndition

eiUzf’(z)isinu Ucosf(z)- -ml

CRITICAL INVESTIGATION OF SUBCLASSES OF FUNCTIONS 91

Izf’(z) zf’ (z) le-ilYet another class is investigated by Gopalakrishna nd Shetiya ]. They nned

this class by SA(e,O,) and f E SA(u,O,D) if and only if f E S with

eiAzf (z)’’f’(’’) isin

cos I - p < p’ (1.5)But we now show that these classes are not new and some of the results of the

above authors are also not new. In fact, we have

THEOREM I.,

(a) S (u,8) S M,M(e,O) M 112(1-8)

(e,); S 282/(1-82(b) Sl(e,8) SI+M,M(c) M(e 8) S*m,S(0,0) m (l+e82)/(l-e282) and M (I+e)8/(I-(282)

(d) SX(,O,) S,s(,) S p/(l-e).,

PROOF. Suppose X zf’(z)/f(z). Then f e S (e,8) satisfies (1.2). Hence,

we have

[(X-n/{2(X-l-(X-) } < . (.61The inequality (1.6) is true if and only if

slx-l 2 > Re{ (X-s) (-I) }.

This last inequality is equivalent to

[X12 {l+e-2eS}. e(l-8) < 0B {x} + (S)By simple manipulations, the last inequality holds if and only if

92 S.K. BAJPAI

,:__(z) 1 1< --CT=" (’)

Hence, if M 2--2, uwith u 0 then f e S (a,8) implies f e SM,M(a,0) mdconversely. This proves (a).

For.proving (b), we let Y {Xeil isinl acosl}l(1-a)cosl and

X zf’ (z) S x(z) Then, as before f e (a,) if and rely if

(1.8)

(In (1.8), if k -0 c, then this class is due to Caplinger and Causey 3). But

simple calculations shoe tha (1.8) is equivalent to

il zf’ (z) isinl-acoslf (z) 282

282--(l---a) cosX 1 <

1-2 1-" (1.9)

Thus, a comparision of (1.9) to (1.1) implies (b) and conversely. The proof for

(c) is similar. For (d) we observe tha f e Sl(cz,p,q) if md only if

ejAzf ’(z) isinl acosX’(z)""’(:t’)"cos), < I--" (1.10)

Again a cmparision of (I.I0) to (I.I) gives (d). Hence the proof of the theorem

is complete.

We note as a consequence of theorem I, that the coefficients estimates

obtained in the Series of ppers (I]-6, 8]-Ol, [-133-27, 29], 30) arecontained in [7]. But some of the coefficients estimates obtained in [I, [6],

ZI2 are not contained in the theorem 1 of [7]. We also note that the classintroduced in 83, is a very general one but appears unnatural in thepresentation. Therefore, next section,we shall consider the natural generalization

of the class S*m,M(,v) by admitting m to be complex. This we do as follows.


,2. The class Sm,M(,,o,t)._

,We define he class Sm,M(,U,a,t)_ to denote the class of f E S with

elUzf’(z) isinu cosu(I-) cosv 0.

The following lenna shows that when u 0, the utmost class of Szynal et al

and he class introduced by Libera and Livingston 1 are obtained.

S*LEMMA I. f E (U,v,a t) if md only if here exists some w(z) regular inm ,ME, w(0) 0 and [0(z) < 1 such that

zf’(z) I + (De-iV B) w(z) (2.2)"-EG)-- -- s ,.o(z)where D (A+S)(1-U)cceV; A+B > 0and

A= _2_ m2+ m(l_2u)+a(l_u)+it_t2B=. 1 +m+ a-i

PROOF. Let us write

L(z) R m a i

g(z) eiVzf’(z) isinv f(z) Uccev f(z)md h(z) (I-) cosy (z).

ll-m-a-itJ. Hence, there exists a (z) which isThen, we have [L(z)[ < I, L(0) =regular in E and which cm be written in the form

94 S.K. BAJPAI

(z) L(.Z,.) .L_(0)I-L(O)L(z)

Clearly, we have 00(0) 0 and l(z) < i. Solving the equation (2.3), we have

L(z) L(z) m-oiL (0) +.(z)_I+L(0) (z)

and this in turn on solving gives the lenma.

Using this lemma, we prove the following theorem which generalizes the

THEOREM 2. For Izl < I, let f(z) z +k=n+ I

kz and f E Sm,M(,v,,t).Also, let qo be a natural number such that qo E o-I, Ko where

(i) If, either Re{Beiv} > O and M(l-)cos[D + 2n Re{BeiV}] _< n2Re{Bei} < 0 md DM(l-)cosv < n2 then we have

or,

(p+ l)n(k-l) 2 ak 12

knp+ I< D2; p i, 2, 3,

2(ii) If Re{BexV} > 0 and (l-)Mcos + 2n Re{BeXV}] > n(q+l)n q-I 2

{ D+nmBe" }I (k-1)2lak 12 n II n


The estimates (2.4) are sharp for all k and the estimates in (2.5) are sharp for

k qn+l, q 1,2,... because equality holds for the functions

f(z)

DZe-kB for B 0l-Bzk)

De-izk/ke for B -0(A)

PROOF. Under the conditions of theorem 2, we have

k0J(Z) I bkZk=n

for which (2.2) is true. Then simple calculations lead to

[ ak(k-l)zk- z+ I {D+B(k-l)ei}akzk3 [ I bkzk]"k’+l k--n+l k-n

(2.7)

Equating the coefficients of zk frcm both sides of (2.7) for k n+l,...,2n,

we have

e (k-l) ak Dbk_I.

This immediately gives us,

2nD2 2 D2I (k-l)21ak 12 < . Ibkl <

k=n+ I k=n+I(2.9)

But, for p > n+l, (2.7) can be written in the form

G(z) H(z) 0(z) (2.10)

where

and

n+pG(z) e:l’V n (k-l)akzk + [k= + I k-n+p+ I dkzk

P kH(z) Dz + [ {D+B(k-l)eiV}akzk=n+ I

Then (2.10) gives us

96 S.K. BAJPAI

n+p2 2, (k-l)[ak[ +

k=p+ 1 k=n+p+ 1

k=n+ 1(k-)

From the hypothesis of (i) of theorem 2, we have

and so

ID+(k-1) BeiVl2_

(k-l) 2

pn (k-1)21ak 12 < D2;k=p+l p >n+l.

Replacing p by pn in (2.12) then we have

(Pl)n(k-1)2]akl2 < D2k=pn+l

f or p-2,3,...

Combining (2.13) with (2.9), we get (i).

If Re{Bei9} > 0 then the functicx%

(k_ll) 2 [i D+ (k_l) Bei 12_ (k_l) 2]is always a decreasing function of k-l.

On using this fact, we get,.by (2.11) for p > n+l.

(k-l) akl < D2+ D+nBei 12 n2k--p+ 1 k=n+ I n (k- I)

Let p 2n. Then (2.14) gives us

(2+l)n(k-l)mlak 12 < D,-----6

2n+lby (2.9),

Under the conditions of (ii) of theorem 2, we intend to prove the following.(q+l)n, (k-l) 2 ak 12k=qn+ 1 < ’nl)’ qKl D+mnBezu 2.m=0 n

(2.zz)

(2. 2)

(2.13)

(2.14)

(2.15)

(2.16)


The relation (2.16) is true for q I and 2, so let us assume (2.16) to be true

for 1,2,...,q-I and let us prove it for q if q < qo + I. By (2.14) we have(q/l)n

2 i72[ lakl2(k-l) < + ID+(k-l)Be1912 (k-l) 2 II 2k=qn+ I k--n+I

D2 + m--q-ll k(m+l)n!n+I { ID+ (k- l) Beiv12(k-l)2 (k-l) 2 l(k-l)2 ak 12

+m:l m

2 2 "(m-l)’m=0

m=0 n

The last inequality also follows by induction. This proves (ii) of theorem 2.

Also, if k > (qo+2)n then+l)n(q+l)n 2 -’(qi qn 2 2} 2]+ [ l)n+l ID+Be19(k-I) -(k-l) akl

(qo+l)n< D2+ I ID+Bei(k_l) 12_(k_I)2> 2 2

k-n+l [ (k-l)2(k-l) ak

m--I k--mn+l (k-l) 2(k-l) 21ak 12

m=0

This proves (iii) of theorem 2. Hence the proof of the theorem is complete.

COROLLARY. From (2.11) we obtain

{(k_l) 2k--n+ I

[D + (k-l)Bei[2}[ak 12 < D2.

obtained in (13-73) are contained in theorem 2 for the different choices of theparameters, m,M,/,9,o and t.

98 S.K. BAJPAI

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