[PDF] COEFFICIENT ESTIMATES FOR BI-CONCAVE FUNCTIONS 1





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COEFFICIENT ESTIMATES FOR BI-CONCAVE FUNCTIONS 1

13 feb 2018 Introduction Preliminaries and DeFinition. The knowledge on bi-concave univalent functions is based on univalent

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

Volume 68, Number 1, Pages 53-60 (2019)

DOI: 10.31801/cfsuasmas.443600

ISSN 1303-5991 E-ISSN 2618-6470Available online: February 13, 2018

http://communications.science.ankara.edu.tr/index.php?series=A1COEFFICIENT ESTIMATES FOR BI-CONCAVE FUNCTIONS

F. MÜGE SAKAR AND H. ÖZLEM GÜNEY

Abstract.In this study,a new classCp;q

()of analytic and bi-concave func- tions were presented in the open unit disc. The coe¢ cients estimates on the ...rst two Taylor-Maclaurin coe¢ cientsja2jandja3jwere found for functions belonging to this class.

1.Introduction, Preliminaries and Definition

The knowledge on bi-concave univalent functions is based on univalent, con- cave and bi-univalent funcions respectively. Therefore, a brief summary of these functions and related references are given in this section. Lets takeCas the complex numbers andRas the set of real numbers. Then

open unit disk can be denoted byDand extended complex plain are denoted byC=C[ f1g. LetAindicate the class of analytic functions in the open unit disk

D=fz2C:jzj<1ggiven in the following form

f(z) =z+1X n=2a nzn:(1.1) All the normalized analytic function classesAwhich are univalent inDare also represented byS. An univalent functionf:D!Cis called to be concave when f(D)is concave, i.e.Cnf(D)is convex. Concave univalent functions have already been studied in detailed by several authors (see [1,2,3,4,7]). A functionf:D!Cis called to be a member of concave univalent functions with an opening angle,2(1;2], at in...nity iffholds the conditions given below: (i)fis analytic inDwhich has normalization conditionf(0) = 0 =f0(0)1.

Additionaly,fful...llsf(1) =1.Received by the editors: February 10, 2017, Accepted: October 13, 2017.

2010Mathematics Subject Classi...cation.30C45.

Key words and phrases.Analytic, bi-concave functions, analytic function, univalent function, bi-univalent function, concave function. c

2018 Ankara University.

Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics.

Communications de la Faculté des Sciences de l"Université d"Ankara-Séries A1 Mathematics and Statistics.

53

54 F. MÜGE SAKAR AND H. ÖZLEM GÜNEY

(ii)fmapsDconformally onto a set whose complement in accordance withCis convex. (iii) The opening angle off(D)at in...nity is equal to or less than,2(1;2]. Lets indicate the class of concave univalent functions of orderbyC(). The analytic characterization for functions inC()are as follows :

For2(1;2]and2[0;1),f2C()if and only if

;8z2D;(1.2) for P f(z) =21 + 12

1 +z1z1zf00(z)f

0(z) and f(0) = 0 =f0(0)1: Especially, for= 0, we can obtain the class of concave univalent functionsC0() which was studied in [3]. The closed setCnf(D)is convex and unbounded forf2C0(),2(1;2].

8f2C()has the Taylor expansion given by the following form

f(z) =z+1X n=2a nzn;jzj<1: For allf2 S, the Koebe1=4theorem [8] con...rms that the image ofDunder all univalent functionf2 Scovers a disk of radius1=4. Hence, eachf2 Ahasf1, which is described by f

1(f(z)) =z(z2D)

and f(f1(w)) =w jwj< r0(f);r0(f)14 Iff(z)is univalent inDandg(w) =f1(w)is univalent infw:jwj<1g, then the functionfbelongs to analytic function is known to be bi-univalent inD. Iff(z) given by (1.1) is bi-univalent, theng=f1can be arranged in the form of Taylor expansion given g(w) =wa2w2+ (2a22a3)w3::: :(1.3) So,f2 Ais called to be bi-univalent inDif each offandf1are univalent inD. Also, a functionfis bi-concave if bothfandf1are concave. Some properties of bi-convex, bi-univalent and bi-starlike function classes have already been investigated by Brannan and Taha [6]. Furthermore, an estimation of ja2jandja3jwas found by Bulut [5] for bi-starlike functions. Our results found for Lets denoteas the class of all bi-univalent functions in the unit diskD. Lewin [10] investigatedand showed thatja2j<1:51for the functionf(z)2. Also, sev- eral researchers obtained the coe¢ cients boundary forja2jandja3jof bi-univalent

COEFFICIENT ESTIMATES FOR BI-CONCAVE FUNCTIONS 55

functions for the some subclasses of the classin references [9,11,12]. In addition, certain subclasses of bi-univalent functions, and also univalent functions consisting of strongly starlike, starlike and convex functions were studied by Brannan and Taha [6] . They investigated bi-convex and bi-starlike functions and also investi- gated some properties of these classes. Now, we de...ne the de...nition of bi-concave functions as follows: De...nition 1.1.The functionf(z)in (1.1) is known to beP C (),(1< 2)if the conditions given below are ful...lled:f2, <21 + 12

1 +z1z1zf00(z)f

0(z) > ;z2Dand0 <1(1.4) and <21 + 12

1w1 +w1wg00(w)g

0(w) > ;w2Dand0 <1:(1.5) where thegis given in (1.3). In the other words,P C ()is the class of bi-concave functions order: We introduce the following subclass of the analytic functions classA, analogously to the de...nition given by Xu et al. [13]. De...nition 1.2.Lets de...ne the functionsp;q:D!Csatisfying the following condition minf<(p(z));<(q(z))g>0 (z2D)andp(0) =q(0) = 1:

In addition letf, in (1.1), be inA. Then,f2 Cp;q

(),(1< 2)if the conditions given in (1.4) and (1.5) are ful...lled:f2 21
+ 12

1 +z1z1zf00(z)f

0(z)

2p(D);(z2D)(1.6)

and 21
+ 12

1w1 +w1wg00(w)g

0(w)

2q(D);(w2D)(1.7)

where thegis given in (1.3).

Remark

If we let

p(z) =1 + (12)z1zand q(z) =1(12)z1 +z(0 <1;z2D) (1.8) in the classCp;q ()then we haveP C The aim of this paper is to estimate the initial coe¢ cients for the bi-concave functions inD.

56 F. MÜGE SAKAR AND H. ÖZLEM GÜNEY

2.Initial Coefficient Boundary forja2jandja3j

The estimation of initial coe¢ cient for bi-concave functions classCp;q ()are presented in this section. Theorem 2.1.If the functionf(z)given by (1.1) is inCp;q ()then ja2j min( r(+ 1)24 +(21)8 [jp0(0)j+jq0(0)j] +(1)232 [jp02+jq02] r(+ 1)2 +(1)16 [jp00(0)j+jq00(0)j]) (2.1) and ja3j min(+ 1)2 +(1)24 [2jp00(0)j+jq00(0)j] (+ 1)24 +(1)48 [jp00(0)j+jq00(0)j] +18 (21)[jp0(0)j+jq0(0)j] +132 (1)2[jp02+jq02] (2.2) Proof.Firstly, we can write the argument inequalities in their equivalent forms as follows: 21
(+ 1)2

1 +z1z1zf00(z)f

0(z) =p(z) (z2D);(2.3) and 21
(+ 1)2

1w1 +w1wg00(w)g

0(w) =q(w) (w2D):(2.4) In addition to, thep(z)andq(w)can be expended to Taylor-Maclaurin series as given below respectively p(z) = 1 +p1z+p2z2+::: and q(w) = 1 +q1w+q2w2+::: :

Now upon equating the coe¢ cients of

21h
(+1)2

1+z1z1zf00(z)f

0(z)i with those of p(z)and the coe¢ cients of21h (+1)2

1w1+w1wg00(w)g

0(w)i with those ofq(w). We can writep(z)andq(w)as follows. p(z) =2(1) (+ 1)2

1 +z1z1zf00(z)f

0(z) = 1 +p1z+p2z2+p3z3+:::(2.5)

COEFFICIENT ESTIMATES FOR BI-CONCAVE FUNCTIONS 57

and q(w) =2(1) (+ 1)2

1w1 +w1wg00(w)g

0(w) = 1 +q1w+q2w2+q3w3+::: : (2.6) Since zf

00(z)f

0(z)=2a2z+ 6a3z2+ 12a4z3+:::1 + 2a2z+ 3a3z2+ 4a4z3+:::= 2a2z+ (6a34a22)z2+:::

and

1 +z1z= 1 + 21X

n=1z n= 1 + 2z+ 2z2+ 2z3+::: we obtain that 21
(+ 1)2

1 +z1z1zf00(z)f

0(z) 2(1) (+ 1)2

1 + (+ 1)z+ (+ 1)z2+:::2a2z(6a34a22)z2+:::

2(1) (1)2 + ((+ 1)2a2)z+ ((+ 1)(6a34a22))z2+::: = 1 +

2[(+ 1)2a2](1)z+2[(+ 1)6a3+ 4a22](1)z2+::: :

Then p

1=2[(+ 1)2a2](1)(2.7)

p

2=2[(+ 1)6a3+ 4a22](1):(2.8)

From (1.3) and (2.4)

wg

00(w)g

0(w)=2a2w+ 6(2a22a3)w212(5a325a2a3+a4)w3+:::12a2w+ 3(2a22a3)w24(5a325a2a3+a4)w3+:::

=2a2w+ (8a226a3)w2:

Then fromq(w)given by (2.6), we have

21
(+ 1)2

1w1 +w1wg00(w)g

0(w) 2(1) (+ 1)2 (+ 1)w+ (+ 1)w2:::1 + 2a2w(8a226a3)w2+::: = 12[(+ 1)2a2](1)w+2[(+ 1)8a22+ 6a3](1)w2+::: :

58 F. MÜGE SAKAR AND H. ÖZLEM GÜNEY

So we can obtainq1andq2as follows

q

1=2[(+ 1)2a2](1)(2.9)

q

2=2[(+ 1)8a22+ 6a3](1):(2.10)

From (2.7) and (2.9) we obtain

p

1=q1(2.11)

a

22=(+ 1)24

(21)8 [p1q1] +(1)232 [p21+q21]: (2.12)

Also, from (2.8) and (2.10) we obtain that

a

22=(1)8

[p2+q2] +4(+ 1)8 :(2.13)

Therefore, we ...nd from the (2.12) and (2.13)

ja2j2(+ 1)24 +(21)8 [jp0(0)j+jq0(0)j]+(1)232 [jp02+jq02] and ja2j2(+ 1)2 +(1)16 [jp00(0)j+jq00(0)j]: So we have the coe¢ cient ofja2jas maintained in (2.1). Now, to obtain the bound on the coe¢ cientja3jwe use (2.8) and (2.10). So we obtain (1)(p2q2) = 24a2224a3:

From (2.13) we ...nd

24a3=(1)(p2q2)+24(+ 1)2

+(1)8 (p2+q2) )a3=(+ 1)2 (1)12 [2p2+q2]:(2.14)

We thus ...nd that

ja3j + 12 +(1)24 (2jp00(0)j+jq00(0)j):

Also from (2.12) we obtain

24a3=(1)(p2q2)+24(+ 1)24

(21)8 (p1q1) +(1)232 (p21+q21) )a3=(+ 1)24 (1)24 (p2q2)18 (21)(p1q1)+132 (1)2(p21+q21): (2.15)

COEFFICIENT ESTIMATES FOR BI-CONCAVE FUNCTIONS 59

We thus ...nd that

ja3j (+ 1)24 +(1)48 (jp00(0)j+jq00(0)j)+18 (21)(jp0(0)j+jq0(0)j)+132 (1)2(jp02+jq02):

So, The the proof of Theorem 2.1 is completed.

3.Conclusion

Ifpandqare chosen in Theorem 2.1 as follows, the following corollary can easily be obtained. p(z) =1 + (12)z1zand q(z) =1(12)z1 +z(0 <1;z2D) Corollary 3.1.Letf(z), in the expansion (1.1) be in the bi-concave function classP C (),(0 <1;1< 2). Then ja2j r(+ 1)2 +(1)2 (1) and ja3j (+ 1)2 +(1)2 (1):

References

[1] Alt¬nkaya, ¸S. and Yalç¬n, S., General Properties of Multivalent Concave Functions Involv-

Sciences,6912(2016), 1533-1540.

[2] Avkhadiev, F. G., Pommerenke, C. and Wirths, K.-J., Sharp inequalities for the coe¢ cient of concave schlicht functions, Comment. Math. Helv.81(2006), 801-807. [3] Avkhadiev F. G. and Wirths, K.-J., Concave schlicht functions with bounded opening angle at in...nity,Lobachevskii J. Math.17(2005), 3-10.

[4] Bayram, H. and Alt¬nkaya, ¸S., General Properties of Concave Functions De...ned by the Gen-

eralized Srivastava-Attiya Operator, Journal of Computational Analysis and Applications,

233(2017), 408-416.

[5] Bulut, S., Coe¢ cient estimates for a class of analytic and bi-univalent functions, Novi Sad J.

Math. 43(2013), no. 2, 59-65.

[6] Brannan, D. A. and Taha, T. S., On some classes of bi-univalent functions ,Studia Univ.

Babes-Bolyai Math.2(1986), no. 31, 70-77.

[7] Cruz, L. and Pommerenke, C., On concave univalent functions ,Complex Var. Elliptic Equ.

52(2007), 153-159.

[8] Duren, P. L., Univalent functions , In. Grundlehren der Mathematischen Wissenschaften, vol.

259, New York: Springer1983.

[9] Frasin, B. A. and Aouf, M. K., New subclasses of bi-univalent functions,Appl. Math. Lett.

24(2011), 1569-1573.

[10] Lewin, M., On a coe¢ cient problem for be univalent functions ,Proc Amer Math. Soc,

18(1967), 63-68.

[11] Srivastava, H. M., Mishra, A. K. and Gochhayat, P., Certain subclasses of analytic and bi-univalent functions ,Appl. Math. Lett.23(2010), 1188-1192.

60 F. MÜGE SAKAR AND H. ÖZLEM GÜNEY

[12] Xu, Q.-H., Xiao, H.-G. and Srivastava, H. M., A certain general subclass of analytic and bi-univalent functions and associated coe¢ cient estimate problems,Appl. Math. Comput.

23(2012), no. 218, 11461-11465.

[13] Xu, Q.-H., Gui, Y.-C. and Srivastava, H. M., Coe¢ cient estimates for a certain subclass of analytic and bi-univalent functions,Appl. Math. Lett.25(2012), 990-994. Current address: F. Müge SAKAR Batman University Faculty of Management and Economics Department of Business Administration 72060 Batman TURKEY

E-mail address:mugesakar@hotmail.com

ORCID Address:http://orcid.org/0000-0002-3884-3957 Current address: H. Özlem GÜNEY Dicle University Faculty of Science Department of Math- ematics 21280 Diyarbak¬r TURKEY

E-mail address:ozlemg@dicle.edu.tr

ORCID Address:http://orcid.org/0000-0002-3010-7795quotesdbs_dbs14.pdfusesText_20
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