On a certain class of univalent functions

J.

ON A CERTAIN CLASS OF UNIVALENT FUNCTIONS

M. K.

S.

M.

1. Introduction

Let f be analytic in a convex domain D. If f satisfies the condition

Re(f'(z»>O (1.1)

for all zED, then it is well known (see [14J, [18J and others) that

f is univalent in D. MacGregor [9J investigated the properties, e. g. , coefficient estimates, radius of convexity etc. for functions f analytic in the unit disc U= {z : Izl <I} having power series representation

=

f(z) =z+ L;anzn

n=2

(1. 2)

and satisfying (1. 1) for

We denote the class of such functions by R. Analogous properties have also been obtained in [9J for analytic functions with initial zero coefficients in (1. 2) and satisfying (1. 1) for z

U. Ezrohi [4J and Martynov [l1J obtained the radius of convexity along with the other properties for the class R of functions f(z) that are analytic and satisfy

Re(f'(z»>a (1. 3)

for O::::;;;a<l,

U. Several other subclasses of R have also been obtained by Caplinger and Causey [2J, Goel [5,6J, MacGregor [10J, Padmanabhan[15J, Shaffer [16J and others.

Let N be the class of functions

=

f(z) =z+ L; anzn

n=k+l

(1. 4)

that are analytic in U. In this paper, we propose a unified approach to the study of various subclasses of univalent functions whose deriva- tives have a positive real part in

Thus, we introduce the class

Received July 28. 1989.

Revised March 2. 1990.

- 145-

146

R,,(a, fJ, A, B) which, for different values of the parameters a, fJ, A, B(O

~a<l, O<fJ~l, -l~A<B~l, O<B~l),

not only gives rise to the classes studied by the above mentioned workers but also gives rise to many new subclasses of univalent functions. Thus we have the following:

DEFINITION.

Let fez) EN. Then fER" (a, fJ, A, B) if the condition

I (B-A)fJ(f,fz;~a)~A(f'(z)-l) ^{1<1 (1.5)}

is satisfied for some a,

and for all z

U. We note that R" (a, fJ, -1, 1) = R" (a, fJ), is the class studied by Mogra [12].

It is easy to check that RI (a, 1, -1, l)is the class Ra studied by Ezrohi [4], Martynov [11] etc. ; RI (0,1, -1,1) =R, RI (0, ~, ^-1,1)., R,,(O, l, -1,1) and R,,(O, 1-0, -1,1), where

give rise to the classes introduced by MacGregor [9, 10], Shaffer [16J, while the cases (a, fJ) = (0, 20

20 1), ^0> ~ and (a, fJ) = (i +~, 1 ~r), O~r

<1 with k=l, A=-l and B=l lead respectively to the classes studied earlier by Goel [5], Padmanabhan [15], Caplinger and Causey [2] etc. ; also k=l, A= -1, B=l and a replacement of a by I-a and fJ by ~ ^{in (1.5)} gives the class introduced by Goel [6].

We further, observe that by special choices of a, fJ, A and B our class R,,(a, fJ, A, B) give rise to the following new subclasses of R:

1-Rl"a=R,,(a, 20

20 1, -1,1)

= {fEN : If'i~;a ^01<0, o>~, O~a<l, ^ZEU},

_ ( A ) - ( -A+Ar Br- A )

2 R" r, ,B -R" Br-A' B-A' A,B

= {fEN: I :;,~;)-=-~ I ^<r, O<r~l, -1~A<B~1, O<B~l, ^ZEU},

3-R",

(A, B) =R,,(a, 1, A, B)

={fEN : I Bf,(z)-[~~z{A-!B)(1-a)J ^{1<1, ZEU},}

147

4-R (A B)=R (-A+A,8-(A-B)a,8 B,8-A A B) k,a,p, k B,8-A ' B - A "

= {lE ^N: I ^{Bf' (z) -} _[~~z{A-! ^{B) (I-a)]} I ^<fJ,

O~a<l, ^0< ,8~1, -l~A<B~l, O<B~l,

U.}

REMARK.

From the class Rk.

(A, B) we note that:

(i) RI,a,p(A, B) =R(a,,8, A, B), is the class of functions fez) given by (1. 2) and satisfying

I Bf'(Z)-[~~(~-=!B)(l-a)] ^{]<,8 (1.6)}

for some a,

and

U.

The class of functions fez) satisfying (1. 6) was introduced and studied by Aouf and Owa [lJ.

(ii) RI,a,p( -1,1) =R(a,,8), is the class of functions fez) given by (1. 2) and satisfying

I _f'{~i~~2a ^{1<,8 (1.7)}

for some a,

and

The class of functions fez) satisfying (1. 7) was introduced and studied by juneja and Mogra

[7J.

From the definition given above it is clear that Ri

,8, A, B) is a subclass of the class of functions whose derivatives have a positive real part in

Also Ri (a, ,8, A, B) eRk(a, ,8', A, B) for

It is easily seen that for fE Rk (a, ,8, A, B), the values fez) lie inside the circle in the right half plane with center at

l-[(B-A)a,8+AJ[(B-A),8+AJ . (B-A),8(l-a) 1-[(B-A),8+A]2 and radius 1-[(B-A),8+AJ2' Further, we assume that fez) ER

(a,,8, A, B). Setting

k-1h( )_ 1-f'(z)

z z - (B-A),8U'(z)-a)+A(j'(z)-l) '

we see that the function h

is analytic in the unit disc U, satisfies

I h

I <1 for

U and h (0) =0. Consequently, by using Schwarz's Lemma [13J, we have h(z) =z4J(z) , where 9(z) is an analytic function in the unit disc

and satisfies 19(z) I <1 for

Thus we get

f' ( ) = 1 + [(B-A)a,8+A]zi9 (z)

z 1+ [(B-A),8+A]zi9 (z)

148

In the present paper, we determine sharp coefficient estimates for functions in Ri (a, /3, A, B), radius of convexity etc. for functions in RI (a./3. A, B). A sufficient condition for a function to be in Ri (a, /3, A, B) has also been obtained. For different values of the parameters a, /3, A,

B(O~a<l, 0</3~1, -l~A<B~l, O<B~l)

our results generalize the corresponding results obtained by Caplinger and Causey [2J, Ezrohi [4J, Goel [5,6J, Kaczmarski [8J, MacGregor [9,10J, Martynov [l1J, Padmanabhan [15J, Shaffer [16J and Mogra [12J.

2. Coefficient estimates

THEOREM

1. If fez) EN is in Ri (a, /3, A, B), then lanl:S;;; (B-A)/3(l-a)

n

for

k=l, 2, .... The bounds are sharp for the functions

.f (z)=(%l+[ -A...,.. (B-A)a/3Jtn-

dt n Jo l-[(B-A)/3+AJtn

for

and zEU.

The proof of the above theorem is similar to that of Clunie [3J, and hence is omitted.

REMARK.

Different values of the parameters a, /3, A, Band k=l in Theorem 1 lead to the coefficient estimates obtained earlier by Caplinger and Causey [2J, Gael [5J, MacGregor [9,10J, Padmanabhan [15J and Mogra [12J.

3. A sufficient condition for a function to be in Ri (a, {J, A, B)

THEOREM

2. Let f(:t) EN. If for some a, /3, A, B(O:S;;;a<l,

(B-A)' -A

I:

(l-A- (B-A)/3)nlan

(B-A)/3(l-a), (3.1)

,,=.1:+1 .

then fez) belongs to RiCa, /3, A, B).

Proof. Suppose (3.1) holds for some a, /3, A, B

(B-=-~)' -l~A<B~l, O<B~l) ^{and that}

00

fez) =z+ I: anzn,

,,=.1:+1

On a certain class of univalent functions

then for z

1

j' (z) -11-1 (B-A)[3(j' (z) -a) +A(j' (z) -1)

:;;; .E+ln lan Ir

{(B-A) [3(l-a) +

.E+I (-A- (B-A)[3)n\an

r

00 00

<

n \an 1- ^{(B-A) [3 (l-a)} +

(-A- (B-A)f3)n \anl

.=1+1 .=1+1

00

=

(l-A-(B-A)[3)nlanl-(B-A)[3(l-a) :;;;0,

'=1+1

by (3. 1). Hence it follows that

149

I (B-A)[3(f'&;~a)~A(j'(Z)-I) ^\<1,

so that jERk(a, [3, A, B). Hence the theorem.

REMARK. Since j(z)ERk(a, (B-!A) ,A,B) impliesjERk(a, [3,A,B)

-A -A

for (B-A) :;;;[3:;;;1, the condition (3.1) for [3= (B-A) , that is, the condition

00

~

nla.\

(I-a)

.=1+1

(3.2)

can also be used as a sufficient condition for a function to be in Rk (a, [3, A, B) for O:;;;a<I, (B~) :;;;[3:;;;1, -l:;;;A<B:;;;l, O<B:;;;l. The condition (3.2) with k=l, a=O, A= -1, B=l may be found in

[19J as a sufficient condition for a function to be in R.

4. Radius of convexity for function in RI

[3, A, B)

Let

denote the class of analytic functions w(z) in U which satisfy the conditions (i) w(O) =0 and (ii) I w(z) I ^{<1 for}

U. We require the following lemmas.

LEMMA 1[17]. Ij wED, then jor

U

I zw' (z) -w(z) I ~ 1 z 1

_1 w(z)

I""

1-lz12 . (4.1)

150

0<,8

LEMMA

2. Let w(z)

Then we have

Re{ (l+[(B-A),8+A]w(~~(¥~[(B-A)a,8+A]w(z»}

-1 { [ ( ) A] () [(B-A)a,8+A]

~ (B-A)2 (l-a)

Re B-A,8+ P z + ^p(z)

-[(B-A),8(l+a) +2A]}

+ ^r

[(B-A),8+A]p(z) -[(B-A)a,8+A]

-11-p(z)

(B-A)2(1-a)2,82(1-r2) Ip(z) I '

h ( ) _l+[(B-A)a,8+A]w(z) -I I

were p z - l+[(B-A),8+A]w(z) , r- z and O~a<l,

~1, -l~A<B~l, O<B~l.

The proof of the above lemma follows from (4.1) immediately. So we omit it.

REMARK.

The transformation

p( )= 1+ [(B-A)a,8+A]w(z) z 1+ [(B-A),8+A]w(z) maps the circle Iw(z) I

onto the circle

I p( )_1-[(B-A)a,8+A][(B-A),8+AJr21

z 1-[(B-A),8+A]2r2

~

(B-A),8(l-a)r

"'" 1-[(B-A),8+A]2r2·

THEOREM

3. Let fez) ERI(a,,8, A, B),

<B~l, O<B~l,

then f is convex in Izl <ro, where ro is the smallest positive root of

V (1 +[(B-A),8+A]) (1 + [(B-A)a,8+A]) (1-[(B-A),8+A]r2) . . (1-[(B-A)a,8+A]r2)

-(1-[(B-A)a,8+A][(B-A),8+A]r2) +[-A- (B-A)a,8] (1-r2)

=0

if

RI and ro is the smallest positive root of

1-2[-A-(B-A)a,8]r+[(B-A),8+A] [(B-A)a,8+A]r2=0 if

where

R = {(l+[(B-A)a,8+A]) (1-[(B-A)a,8+A]r2) }!

o (l+[(B-A),8+A]) (l-[(B-A),8+A]r2)

and

R - 1+ [(B-A)a.B+AJr Izl =r<l.

1- l+[(B-A).B+AJr ' All the above estimates are sharp.

151

Prooj. Since jER1(a,.B, A, B), we have by Schwarz's Lemma [13J j'

= 11~[[~~-=-1)1:~~~» , ^(4.2)

where wED. Differentiating (4.2) logarithmically, we get l+ zj"(z) l-(B-A).B(l-a)·

j'

o

{I ^{+ [(B-A).B+} _AJw(z)~i~ [(B-A)a.B+AJw(z»}· (4.3) An application of Lemma 2 to the above equation gives

{ zj" (z) } 1

Re 1+ j'(z) ~ (B-A).B(l-a) .

o

[Re{[(B-A).B+AJ p(z) + [(B-A)a.B+AJ } p(z)

r²1

[(B-A),8+AJp(z) - A - (B-A)a.B12-ll- p(z)

2 (1-r

I p(z) I

- A - (B-A)a.B

+2 (B-A).B(l-a) , (4.4)

h () _l+[(B-A)a.B+AJw(z) (

were p z - 1+ [(B-A).B+AJw(z) . Setting p z) =a+~+i7), R2=(a+~)2+7)2 where a= 1-[(B-A)a.B+AJ[(B-A).B+AJr 2

, 1-[(B-A).B+AJ2r2 and

denoting the expression on the right hand side of (4.4) by

7), we get

_ - A - (B-A)a.B S(~,

^-2 (B-A).B(l-a)

+ (B-A).B(l-a) 1 [(B-A).B+AJ(a+~)+[(B-A)a.B+AJ·

. (a+~)R-2 1-[(B-;:~~~+AJ2r2 (d2_~2_7)2)R-1J, ^(4.5) ,,,here d- (B-A),8(l-a)r DOff . . (4) ·11

,y

-1-[(B-A),8+A]2r2.

erenhatmg .5 parha y w.r.

152

t.1j,

we get

where

M. K. Aouf, S. Owa and M. Obradovic

(4.6)

It is easy to check that

and so (4.6) gives that the minimum of

1j) inside the disc

is attained on the diameter 1j=0. On putting 1j=0 in (4.5), we obtain

_ _ - A - (B-A)af3 U(R) -S(~, 0) ~2 (B-A)f3(l-a) +

1 [([(B-A)f3+A] + 1-[(B-A)f3+ A ]2r2)R

(B-A)f3(l-a) 1-r2

+ ⁽¹⁺ [(B-A)af3+A])(1-[(B-A)af3+A]r2) R-l 1-r

-2 1-[(B-A)f3+A ]2r2]

a 1-r

where

and

Thus the absolute minimum of U(R) in (0, 00) is attained at

R = I

(4.7) o -V ⁽¹ + [(B--'-A)f3+A]) (1- [(B-A)f3+A]r2)

and the value of this minimum is 2

U(Ro) (B-A)f3(l-a) (1-r2)

{V (1 + [(B-A)f3+A]) (1 + [(B-A)af3+A]) (1- [(B-A)f3 +A]r2(1-[(B-A)af3+A]r2) - (1- [(B-A)af3+A]·

. [(B-A)f3+A]r2) +[ - A - (B-A)af3] (1-r2)}. (4.8)

It is easily seen that Ro<a+d, but R o is not always greater than

a-d. In such a case whf'n Roft [a-d, a+d], the minimum of U(R)

on the segment [a-d, a+d] is attained at R1=a-d since U(R)

increases with R on this segment. The value of this minimum equals

On a certain class of univalent functions

U(R

= ^U(a-d) =

1-2[-A-(B-A)aPJr+[(B-A)P+AJ[(B-A)ap+AJr2 (1 + [(B-A)p+AJr) (1 + [(B-A)ap+AJr)

153

(4.9) It follows from what has been said that the bound ro of convexity for the class RI

P, A, B) is determined either from the equation U(Ro)

=0 or from the equation U(R

=0. Also, _U(Ro) = U(R

for such values of

and

for which Ro=R

From (4.4), (4.8) and (4.9), we have Re{1 + zf" (z) }

I'(z)

(B-A)P(I~a) ^(l-r2) {v' (1+[(B-A)P+AJ) (1+[(B-A)ap

+AJ) (1-[(B-A)p+AJr2) (1-[(B-A)ap+AJr2)- (1-[(B- A)ap+AJ[(B-A)p+AJr2) +[ - A - (B-A)apJ (l-r)}

;;;:;.

if R o;;;:;' ^{RI (4. 10)}

1-2[-A-(B-A)apJr+[(B-A)P+AJ[(B-A)ap+AJr (1 + [(B-A) p+AJr) (1 + [(B-A)ap+AJr)

if

Now the theorem follows easily from (4.10). The functions given by I'(z) = 1-[(B-A)ap+AJz

1-[B-A)P+AJz '

I' ^{(z) =} 1-[1+ (B-A)ap+AJbz+[(B-A)ap+AJ z2 1-[1 + (B-A)P+AJbz+[(B-A)P+AJ z2 ' where b is determined by the relation

1-[1+ (B-A)ap+AJbr+[(B-A)ap+AJr R 1-[1 + (B-A)p+AJbr+ [(B-A)fi+AJr2

show that the results obtained in the theorem are sharp.

REMARK.

Taking different values of the parameters a, P, A,

1,

in Theorem 3, we get the radii

of convexity for functions in different classes obtained earlier by

Caplinger and Causey [2J, Ezrohi [4J, Goel [5,6J, Kaczmarski [8J,

MacGregor [9,10J, Martynov [l1J, Padmanabhan [15J, Shaffer [16J,

154

Aouf and Owa [lJ, Juneja and Mogra [7J and Mogra [12J.

Acknowledgements. The authors express their sincere thanks to the referee for his helpful comments and suggestions.

References

1- M.

Aouf and S. Owa, On a class of univalent functions, (Submitted).

2. T. R. CapliQger and W. P. Causey, A class of univalent functions, Proc.

Amer. Math.

(2) 39(1973), 357-361-

3. J. Clunie, On moromorphic schlicht functions, J. London Math. Soc. 34 (1959), 215-216.

4.

T. G. Ezrohi, Certain estimates in special classes of univalent functions in the unit circle Izl<l, Dopovidi Akad. Nauk Ukrain RSR (1965), 984-

5. R. M. Goel, A class of univalent functions whose derivatives have positive real part in the unit disc, Nieuw Arch. Wisk. (3) 15 (1967), 55-63.

6. R. M. Goel, A class of univalent functions with fixed second coefficients, J. Math. Sci. 4(1969), 85-92.

7. O. P. Juneja and M. L. Mogra, A class of univalent functions, Bull. Sci.

Math., 2c serie, 103(1979), 435-447.

8. J. Kaczmarski, On the radius of convexity for certain regular functions, Comment. Math. Warszawa 17(1974), 745-748.

9. T. H. MacGregor, Functions whose derivatives have positive real part, Trans.

Amer. Math.

104(1962), 532-537.

10. T. H. MacGregor, A class of univalent functions, Proc. Amer. Math. Soc.

15(1964), 311-317.

11. Ju. A. Martynov,' Uber geometrische Eigneschaften der Bogen der Niveaul·

inien bei Schlichten Knformen Abbildungen, Trudy Tomsk, gosuderst Univ. V. V. Kulbysev 210 ser. meh. mat., Voprosy geom. Teor. Funkcil 6(1969), 53-61.

12. M. L. Mogra, On a class of univalent functions whose derivatives have a positive real part, Riv. Mat. Univ. Parma (4) 7(1981), 163-172.

13. Z. Nehari, Conformal Mapping, McGraw Hill Book Co., Inc. (1952).

14. K. Noshiro, On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. Soc. I Japan 2 (1934-35), 129-155.

15. K. S. Padmanabhan, On a certain class of functions whose derivatives have a positive real part in the unit disc, Ann. Polon. Math. 23(1970/71), 73-81-

16. D. B. Shaffer, On bounds for the derivatives of analytic functions, Proc.

Amer. Math.

37(1973), 517-520.

On a certain class of univalent functions

17. V. Singh and R. M. Gael,

J. Math. Soc. Japan 23(1971), 323-339.

18. S. E. Warschawski,

at

Trans. Amer. Math. Soc. 38(1935), 310-390.

19. Zeller, Theorie der limitierungs verfahren, Berlin 1958.

University of Mansoura Faculty of Science Mansoura, Egypt, Kinki University Higashi-Qsaka, Osaka 577, Japan and

Faculty of Technology and Metallurgy 4, Karnegieva Street

11000 Belgrade, Yugoslavia