Vietnam 1 Math (2014) 42:53-62 DOI 10 1007/S10013-013-0032-4
Certain Subclasses of Close-to-Convex Functions
Som P. Goyal • Onkar Singh
Received: 23 February 2013 / Accepted: 12 July 2013 / Published onhne: 13 August 2013
® Vietnam Academy of Science and Technology (VAST) and Spnnger Science-I-Busmess Media Smgapore 2013
Abstract In the present paper certain stibclasses of close-to-convex functions are investi- gated. In particular, we obtain an estimate for the Fekete-Szego functional for functions belonging to our class, coefficient estimates and a sufficient condition. The results presented here would provide extensions of those given in some earlier works.
Keywords Analytic functions • Starlike function • Close-to-convex functions • Subordinations
Mathematics Subject Classiacation (2010) 30C45
1 Inti^uction
Let A denote the class of functions / normalized by
f(z)=z + f]a.f (I)
H = 2
which are analytic in the open unit disk U^{zeC:\z\< 1}. Sakaguchi [10] introduced a class of functions f sA called starlike functions with respect to symmetric points satisfying the condition
Re '^'^'^ >0.
fiz)-fi-z)
These functions are close-to-convex functions. This can be easily seen by showing that the function ifiz) - fi-z))/2 is a starlike function in U. Motivated by the class of starlike
S.P. Goyal (El) • O. Singh
Department of Mathematics, University of Rajasihan. Jaipur 302004, India e-mail' [email protected]
O Singh
e-mail [email protected]
S Springer
S.P. Goyal, O. Singh functions with respect to symmetric points, Gao and Zhou [3] discussed a class K., of close-
K univalent functions. A function feK^ii it satisfies the following inequality:
Ref '"f'^'^ ) < 0 izeU)
\giz)gi-z))
for some funcUon g e S ' (1 /2), 5* (a) denotes the subclass of functions which are starlike of order a in W ( 0 < a < l).Theideahereistoreplacetheaverageof/{2) a n d - / ( - r ) bythe corresponding product —giz)gi—z), and the factor z is included to normalize the expression, so that —z'f'iz)lgiz)g(—z) takes the value 1 at z = 0. To make the function univalent, it is further assumed that g is starlike of order 1/2 so that the function —giz)gi'-z)lz is starlike, which in turn implies the close-to-convexity of / . For some recent works on the problem, see [11-14]. Instead of requiring the quantity —z^f'iz)/giz)gi—z) to lie in the right-half plane, we can consider more general regions. This could be done via subordination between analytic functions.
Let / and g be analytic in U. Then / is subordinate to g, written as f < g or fiz) -<
giz) (E e U), if there is an analytic function wiz), with wiO) — 0 and |w(z)| < 1, such that fiz) = gi^iz)). In particular, if g is univalent in U, then / is subordinate to g, if /(O) —giO) and fiU) c giU). For further information see [2], In terms of subordination, a general class Ksi<p) is introduced in the following defimtion due to Wang et al. [12].
Definition 1 For a function ip with positive real part, the class Ksi<(>) consists of functions f eA satisfying
-z'f'iz) g(z)gi~z)
for some function g e 5 ' ( l / 2 ) .<<piz) izeU)
For (piz) ^ [LT (0 < / < 1)- we get a class KAv) studied by Kowalczyk and Les- Bomba [8]
Recently Goyal et al. [5] have introduced and studied a new class Ks{A,B\u,v) of analytic functions related to starlike functions, as follows:
Definition 2 If / e A we say that / e K,iA, B; u, v), if there exists a function g e S'(l/2) such that
uvz^f'iz) \-^Az giuz)givz) l-f-fiz' w h e r e - I < f l < A < 1; M, u e C * : = C \ { 0 } ; |H| < 1 and |v| < 1.
Also, Wang and Chen [11] introduced and investigated a class K, ik,A,B) satisfying the subordination condition
z^f'iz)+Xz^f"iz) \-\-Az ^ ,,^
-giz)gi-z) ^TTTz ^'^^^'
w h e r e O < ^ < l , - l < B < A < 1 andg € 5*(l/2).Motivated essentially by the aforementioned classes K^i<p) and K„ik,A,B), we intro- duce a more generalized class K,ik,iJ., <p) of analytic functions and obtain some interesting results.
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Definition 3 For a function tp with positive real part, a function f eAi^ said to be m the class K,ik,p, (p) if it satisfies the following subordination condition:
z^f'iz) + z^f"iz)ik -p + 2Xp) + X^z*f"'iz) , ^
~giz)gi-z) where 0 < ^ < A < l a n d g e 5*(l/2).
Obviously for ;Ci = 0 and ^(z) = | ± ^ ( - 1 < fi < A < 1) in (2), we get the class K^ik.A.B) and for A, = ^ = 0, we getthe'^class K^i(p).
Further, if ip(z) ^ j i ^ , then we get the class K,ik. p.j\, B) defined below.
Defimtion 4 A function / e ^ is said to be in the class K^k.p.A.B) if it satisfies the subordination condition (2) with ipiz) replaced by | ± | t . Thus we have
K^{X,p,A,B)
[f.f^A z V ( z ) + z'f"iz)ik -11 + 2kii) + kpz^f'iz) \+Az\
where 0 < ^ < ; . < 1 , - 1 < e < A < 1 a n d g e 5*(l/2).
After a simple calculation, we see that condition in (3) is equivalent to
\z-fU) + z'f"(.z)a-id + Hp)+!.id.z'f"{z) A I «(z)S(-z)
, I -BfaVfe) + z'f'bKx - (1 + ikid) + iidz'f'iz)} I |_^
I giz)g(-z) I
In our proposed investigation of the class Ki{X,ii.'fi).we need the following lertunas:
Lemma 1 (see [3]) Let g e 5 ' ( l / 2 ) and
g(z)=z + f]b,z". (5)
2
z + Y,B,_,_,z". (6)
where
IB2.-1I = |2 f'!.-i-2* 2 * 2, -2 + • • • + 2 ( - l ) " * . - , i ; . + , + ( - I ) " + ' 6 ; |
< 1 ( / . = 2 , 3 , 4 , . . . ) . (7) Lemma 2 (see [9]) Ut f(z) = l + E £ . i « z " de analytic in U and g (z) = 1 + Y.T.1 * z '
be analytic and convex in U.lf f < g. then
| c , | < | r f , | ( t e N = l , 2 . . . . ) .
© Sprmger
56 S P Goyal. O. Singh Recently several authors [1,4-6, 8,11-14] have investigated various subclasses of close- to-convex functions and established a number of interesting results to enrich the theory on these functions.
In the present paper, we aim at proving such results as coefficient estimates and a sharp estimate for the Fekete-Szegb functional for functions belonging to the class Ksik,fi, tp).
We also establish a sufficient condition for fiinctions to belong to the investigated class.
2 Coefficient Estimates
Theorem 1 Let the function f € K,iX, ii, tp), and 0 < ix < X < I, then nlf'm
2n[l-¥i2n-l)iX~p-\-2nXp)]
and
n\<p'iO)\ + l
• (2w 4-1 )[1 + 2n[X -ii-^i2n + l)kii]]"
Proof From the definition of Ksik,p,<p), we know that there exists a function with positive real part
p { z ) = H - £ ] ? „ ; " izeU)
^ . z - / (z) + z V ( z ) a -11 + 2Xp) + X/LizV (z) zF'iz) , , P(z) = T-——^ = - r r r — •< <piz),
-g(z)gi-z) Giz) where Fiz) - zf'iz) -\- z^f"iz)ik -ii + 2Xp) + kp.z'^f'iz).
By Lemma 2, we know that
l A l < k ' ( 0 ) | O e N ) . ( At the same time, by Lemma 1, we have
^ . V -giz)gi-z) , \ ^ D « ,, Giz) = = z-\-2_^B2„_jz (1 and
| B 2 n - i l < l ( / i e N \ ( l } ) . {1 Putting values of / ( z ) , giz) and piz) in (8), we get
(1 + Piz + p2Z^ + '•){z + fisz' + Biz^ + Bjz'' + - • •)
^z{\-\- 2a2Z + 3a-iZ^ + . . ) + {X _ ^ -|- 2kp.)z^{2a2 -f ^a^z -|- 12a4Z^ -\-..-) + kuz^(6^3 + 24a4z -I- 60(152^ H )•
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Certain Subclasses of Close-io-Convex Functions
Comparing the coefficients of z ^ and 2""+' on both sides, we get P2.-1 + P2,-lB, + P2n-sBs + • - - + Pi B>,-I
= 2/i[l + (1 - ^ + 2Xtd){2n -l) + Xp(2n-l)(2n- 2)]o2,
P2» + P2.-2B3 + P2»-4B5 + • • • + P2B2—I + S2„+l
= (2n + 1)[1 + 2 n ( i - fl + 2 i / t ) + 2tii(i(2/i - l)]a2.+]•
From (9), (11), (12), and (13), we get
2n[l + ( i - id. + 2Xid){2n - 1) + Xp(ln - l)(2n - 2)]|02.| < /i|»'(0)|
and
(2n + \)[\+2n(X-ii + 2Xn) + 2nXp.(2n - l)]|<l2,+i I <n\4>'{0)\ + 1 so that
n|»>'(0)l 102.1 <
2;i[H-(2n - l ) ( i - p + 2iii(i)]
"ly'(ll)l + l
|a2.+il_ ( 2 „ + ! ) [ ! + 2 n ( i - p + ( 2 i i + l ) i f J ) ] ' This completes the proof of the theorem.
Setting ft - 0 and ^(z) = { ^ in the above dieorem, we get Corollary 1 Iftheftinctlon f G KAX, A. B) andO<X < 1, then
n(A-B) I02.I; - 2 t l [ l + ( 2 n - l ) i l and
| . . . . i l < "<^-^> + ^ in em ' ' " + " - (2/1 + l ) [ l + 2 / i X ] which is the result obtained earlier by Wang and Chen [11].
Corollary 2 / / w e sef A = 1 ant/ B = - 1 in the above corollary, we get
\aj< ineU) ' " ' - [ l - K / i - l W
which on setting X = 0 reduces to the result due to Gao and Zhou [3].
Further if we set X = /t = 0 in Theorem I, it reduces to the result obtained earlier by Xu etal. [14].
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S P Goyal, O, Singh 3 Fekete—Szego Inequality
We assume that the function ipiz) is an analytic function with positive real part that maps the unit disk U onto a stariike region which is symmetric with respect to the real axis and is normalized by tpiO) = I and ^'(0) > 0. In such case, the function tp has an expansion of the form (piz) = \ + DiZ + T>2Z^ + • • • , D , > 0
Theorem 2 (Fekete-Szegb inequality) For a function fiz) = z-\-a2Z^ + ajz^ -\ belong- ing to the class K^iX, ii,<p). the following sharp estimate holds:
\ai-&al\
3(1 +2X-2p-\-(,kiJ.) ( Dy -Fmaxi 3(1 -\-2k-2p + (ikp)'
\3i\+2k-2iJ. + 6kiJ.) " Ai\+k-ix + 2kp)^\) ^ Proof Since the function / e Ksik.p., tp), there exists a normalized analytic funcfion g e 5*{l/2) such that
z^nz) + z^f"iz)ik-p + 2kp) + kixz^f"'iz) 7~n—^ ^ ^f^'-
-giz)gi-z)
By using the definition of subordination between analytic functions, we find a function io(z) analyfic mW, normalized by u)(0) —0 satisfying |w;(z)| < 1 and
z^f'iz) + z\f"iz)ik - M + 2AM) + Xpz^f'iz) . , ,,
———^ = tf>(w(£))- 14 -giz)gi-z) ^ '
By wnung wiz) — w\z + ui2Z^ + w^z^ -\ , we see that
<p{wiz)) = i-\-Djwa+ (DiW2 + D2W^)z'' + • • - (15) Also by writing giz) = z-\- g2Z^ + giZ^ + • • •, a calculation shows that
- g ( z ) g ( - z ) , 3, J
^ z -I- (2^3 ~ gi)z' + •••
and therefore
, r ' , - - [ i - ( 2 g i - r f ) z ' + • • • ] • g(z)g(-z) z"- ^ ^ ' -"
Using this and (1) in (]4), we get
- ( z ' / ' ( z ) + zV"(z)(X - /! + 2Xfi) + X,tz'f"'(.z)) S(z)g(-Z)
= l + 2 a 2 Z ( l + i - f i + 2X(i)
+ Z^[3<13(1 + 2 i - 2,i + 61M) -2gi + sl]-... (16) fi Springer
Certain Subclasses of Close-to-Convex F Using (14), (15) and (16), we see that
2o2(l -t-X-p + 2kii.) = DxWu
3^3(1 +2k-2p-ir 6Xp) = D,W2 + D2W^ -\-2gj -g\.
This shows that
ai — i5a, = 1 fi3
3(1 -¥2X-2ii-v6kp) V 2 J 3( 1 -I-2A - 2 / / -|-6kp)
\i\.i.k-p + 2kii)y ' I J '
By using the estimate ([7, Inequality 7, p.lO])I w i - t w f l ^ m a x f L l f l l ( r e C )
for an analytic function w with uiiO) = 0 and |u!(z)| < 1 which is sharp for the functions wiz) — z^ or wiz) = z, the desired result follows upon using the estimate that ]g3 - ^ 1 < | for analytic function giz) — z + g2Z^ + g3Z^ + • which is starlike of order \, therefore,
I T l 1
£23 - S a i l <
I -^ ^'- 3il-i-2k-2ii + 6kp) / n.
-\-max' 3il+2k-2ii + 6kii)
I D2 SDJ 13(1 +2k-2p-h6kp) 4i\-\-k-p + Ikpf- \)
and the proof of Theorem 2 is complete. D For p=0. Theorem 2 reduces to
Corollary 3 For a function fiz) =z-\-a2z}+azz^-\ belonging to the class K.ik.O. <p), the following sharp estimate holds:
|a3 - 5a21 < ^^^ ^ 2;,) + """"^l^Sd -H 2).)' 13(14- 2X) 4( 1 -I- X)'- \)' Setting A = 0 in the above corollary, we get
Corollary 4 For a function fiz) = z-\- a2Z--\-a3Z^ + ••• belonging to the class K^tp), the following sharp estimate holds:
\a3-5al\ < - -h max)^—. | — - _ | J which is the same result obtained by Cho et al. [1].
Setting ^ = 0, the above theorem gives the sharp estimate for the third coeilicient of the function in K,{k,p,<p).
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60 S P. Goyal, O. Singh CorollaryS Forafunctionfiz)^z-l-a2z'^ + aiZ^-\- • belonging to the class KsiX. p., (p), the following sharp estimate holds:
1 D, / , |D2l\
n. i < 1 maxi 1, — ^ . ' " - 3 ( 1 + 2 1 - 2 ( 1 + 61(1) 3(1 + 21 - 2)i + 61(1) V D, / For M = 0 the above Corollary 5 reduces to
Corollary 6 fot- a function fiz) ^z + a2Z^ + a3Z^+ ••• belonging to the class KAX,0,<p), the following sharp estimate holds:
' 3 ( 1 + 2 1 ) 3 ( 1 + 2 1 )
For X ^ 0 Corollary 6 gives the sharp estimate for the diird coefficient in K^if) given earlier by Cho et al [1],
1 Di / , \P>2\\
Again if we let .5 -*• co in Theorem 2, it gives the sharp estimate for the second coefficient of the function in KsiX, IL, ip).
Corollary 7 For a function fiz)^z-^ atzr + a^z^ -f- • belonging to the class K,iX,p.,(p), the following sharp estimate holds:
' ' - 2il-\-k-iJ, + 2kiJ.) Setting 11 = 0 Corollary 7 reduces to
CorollaryS For a function fiz)—z-{-a2z'^-i-a^z^-\- • belonging to the class Kiik,0,ip), the following sharp estimate holds:
For A. — 0 in Corollary 8, we get the sharp estimate for the second coefficient in K^ iip) given by Cho et al. [1]:
Also for A — fi — 0 and tpiz) = | r r , Theorem 2 reduces to the corresponding Theorem 2 in [3]
4 Sufficient Condition
Theorems Let g be the function given by (5) and —I < B < A < 1. If an analytic function f e A defined by (1) satisfies the inequality
(1 + \B\)f^n[l -\-in- Dik -11 + nkp)]\a„\ + (l + \A\) J ^ IB^^., | <A-B (!7)
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Certain Subclasses of Close-to-Convex F
and forn =2,3,... the coefficients B2„-i are given by il), then f e K,ik. p.. A , B).
Proof We set for F and G given by (8) and (6), respectively. Now
-\-^^^^^^^-^-B[zf'iz) + z^f"iz)iX-p + 2Xp) + Xpz^f"'iz)]\
^\j^""',z''[\ + in - l)iX - p + nXp)]-f^B2.,-iZ-''-'\
- (A - B)z - BY^na„z"[l + in - 1)(A - p + nXp)] + A ^ B . ^ - i z ^ " "
I 1=2 „=2
Thus, we have
A i ^ n [ l + ( n - l ) ( l - M + til»t)]l<i.||zr+X;iB2.-illzP"-'
,7=2 n=2
-\(A- B)\z\ -tB\J^n[l + (n- 1)(1 - (t + nlft)]|o,||zl"
L "=2 - i ' i i X ; i « 2 . - i i i z i ^ - '
n=2 J
= (i + ifii)^(i[i + (»i-i)(i-/i+iii)i)]io.iizr
n=2
-(A- «)|zl + (1 + \Ai)Y,lB2,_Mz\"-'
<\-(A-B) + {t + lBl)f^n[l + (n-l)l,X-li. + nlp)]K L «=2
+(i+i/ii)x;is2.-ii izi<o.
From the above calculation, we obtain A < 0. Thus, we have
|z/'(z) + zV'"(z)(l - (i + 21p) + iftzV'Cz) - ^^'''^'^ ^ '
< Us[z/'(z) + zV"(z)(J- -fi + 21(1) + i/t;V"'(z)] - '''''''^*'~'^' which IS equivalent to inequality (4). Thus / e ^",(1, ft,/I, B).
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S.P. Goyal, O. Singh Setting /I — 0 in the above theorem, we get
Corollary 9 Let g be the function given by i5) and-I < B -c A < \. If an analytic function f mU defined fey (1) miisfies the inequality
{\ + \B\)Y,n[\-X + nkMa„\ + (\ + \A\)f^\B2.-i\<A-ti
„=2 "=2
and for n =2.3,... the coefficients B2„-} are given by il), then f eK^ik.A.B).
Further setting X = 0 in the above corollary, we get
CoroUary 10 Let g be the function given by (5) and -I <B <A<[. If an analytic func- tion f in U defined byiV) satisfies the inequality
(1 + \B\) Y,n\a„\ + {\ + | A | ) ^ l f i 2 „ - i l <A-B r:=2 n=2
andforn = 2,3,...the coefficients B2n-\ are given by il). then f e K,iA,B).
By setting A - 1 - 2 ) / , 6 = - l in Corollary 10, we get the result obtained by Kowalczyk andBomba [8].
Acknowledgement The authors are thankful lo CSIR, New Delhi, India for awarding Emeritus Scientist- ship lo S.P.G and JRF lo O.S.. under .scheme number 21(084)/10/EMR-II
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