Let f(z)=z+∑n=2∞anzn be analytic in the unit disk with the second coefficient a2 satisfying a2=2b, 0≤b≤1. Sharp radius of Janowski starlikeness is obtained for functions f whose nth coefficient satisfies an≤cn+d(c,d≥0) or an≤c/n(c>0andn≥3). Other radius constants are also obtained for these functions, and connections with earlier results are made.
1. Introduction
Let A denote the class of analytic functions f defined in the open unit disk D≔{z∈C:|z|<1}, normalized by f(0)=0=f′(0)-1, and let S denote its subclass consisting of univalent functions. If f(z)=z+∑n=2∞anzn∈S, de Branges [1] obtained the sharp coefficient bound that |an|≤n(n≥2). However, the inequality |an|≤n, n≥2, is not sufficient for f to be univalent; for example, f(z)=z+2z2 is clearly not a member of S.
Several subclasses of S possess a similar coefficient bound. For instance, the nth coefficients of starlike functions, convex functions in the direction of imaginary axis, and close-to-convex functions satisfy |an|≤n(n≥2) [2–4]. Other examples include functions which are convex, starlike of order 1/2, and starlike with respect to symmetric points. The nth coefficients of these functions satisfy |an|≤1(n≥2) [5–7]. The nth coefficient of close-to-convex functions with argument β satisfies |an|≤1+(n-1)cosβ [8], and the coefficients of uniformly starlike functions are bounded by 2/n [9], while |an|≤1/n [10] for uniformly convex functions. Simple examples show that these bounds are not sufficient to characterize the geometric properties of the classes of functions.
In the sequel, we will assume that f∈A has the Taylor expansion of the form f(z)=z+∑n=2∞anzn. Gavrilov [11] showed that the radius of univalence for functions f∈A satisfying |an|≤n(n≥2) is the real root r0≃0.164 of the equation 2(1-r)3-(1+r)=0, and the result is sharp for f(z)=2z-z/(1-z)2. Gavrilov also proved that the radius of univalence for functions f∈A satisfying the coefficient bound |an|≤M(n≥2) is 1-M/(1+M). The condition |an|≤M clearly holds for functions f∈A satisfying |f(z)|≤M, and for these functions, Landau [12] proved that the radius of univalence is M-M2-1. In fact, Yamashita [13] showed that the radius of univalence obtained by Gavrilov [11] is also the radius of starlikeness for functions f∈A satisfying |an|≤n or |an|≤M. Additionally, Yamashita [13] determined that the radius of convexity for functions f∈A satisfying |an|≤n is the real root r0≃0.090 of the equation 2(1-r)4-(1+4r+r2)=0, while the radius of convexity for functions f∈A satisfying |an|≤M is the real root of
(1)(M+1)(1-r)3-M(1+r)=0.
Recently, Kalaj et al. [14] obtained the radii of univalence, starlikeness, and convexity for harmonic mappings satisfying certain coefficient inequalities.
For two analytic functions f and g, the function f is subordinate to g, denoted by f≺g, if there is an analytic self-map w of D with w(0)=0 satisfying f(z)=g(w(z)). If g is univalent, then f≺g is equivalent to f(0)=g(0) and f(D)⊆g(D).
For β∈R∖{1}, α≥0, the class L(α,β) consists of functions f∈A satisfying
(2)αz2f′′(z)f(z)+zf′(z)f(z)≺1+(1-2β)z1-z.
Denote by L0(α,β) its subclass consisting of functions f∈A satisfying
(3)|αz2f′′(z)f(z)+zf′(z)f(z)-1|≤|1-β|(β∈R∖{1},α≥0).
These classes were investigated in [15–24].
For β<1, the class L(0,β) is the class of starlike functions of order β, while, for the case β>1, the class was studied in [25–28].
The class ST[A,B] of Janowski starlike functions [29] consists of f∈A satisfying the subordination
(4)zf′(z)f(z)≺1+Az1+Bz(-1≤B<A≤1).
Certain well-known subclasses of starlike functions are special cases of ST[A,B] for appropriate choices of the parameters A and B. For example, for 0≤β<1, ST(β)≔ST[1-2β,-1] is the familiar class of starlike functions of order β. Denote by STβ the class STβ≔L0(0,β)=ST[1-β,0]. Janowski [29] obtained the sharp radius of convexity for ST[A,B].
This paper studies the class Ab consisting of functions f(z)=z+∑n=2∞anzn, (|a2|=2b,0≤b≤1), in the disk D. The subclass of univalent functions in Ab have been studied in [30–33]. In [33], Ravichandran obtained sharp radii of starlikeness and convexity of order α for functions f∈Ab satisfying |an|≤n or |an|≤M, n≥3. The author also obtained the radius of uniform convexity and parabolic starlikeness for functions f∈Ab satisfying |an|≤n, n≥3.
This paper finds radius constants for functions f(z)=z+∑n=2∞anzn∈Ab satisfying either |an|≤cn+d(c,d≥0) or |an|≤c/n(c>0,n≥3). In the next section, sharp L(α,β)-radius and ST[A,B]-radius are derived for these classes. Several known radius constants are shown to be special cases of the results obtained.
2. Radius Constants
A sufficient condition for functions f∈A to belong to the class L(α,β) is given in the following lemma.
Lemma 1 (see [24, 34]).
Let β∈R∖{1} and α≥0. If f(z)=z+∑n=2∞anzn∈A satisfies the inequality
(5)∑n=2∞(αn2+(1-α)n-β)|an|≤|1-β|,
then f∈L(α,β).
Making use of this lemma, the sharp L(α,β)-radius is obtained for f∈Ab satisfying the coefficient inequality |an|≤cn+d.
Theorem 2.
Let β∈R∖{1}, 6α+3-β≥0, and α≥0. The L(α,β)-radius for f(z)=z+∑n=2∞anzn∈Ab satisfying the coefficient inequality |an|≤cn+d, c,d≥0, n≥3, is the real root in (0,1) of the equation
(6)((c+d)(1-β)+|1-β|+(2α+2-β)(2(c-b)+d)r)(1-r)4=cα(1+4r+r2)+((1-α)c+αd)(1-r2)+((1-α)d-βc)(1-r)2-βd(1-r)3.
For β<1, this number is also the L0(α,β)-radius of f∈Ab. The results are sharp.
Proof.
The number r0 is the L(α,β)-radius for f∈Ab if and only if f(r0z)/r0∈L(α,β). Therefore, by Lemma 1, it is sufficient to verify the inequality
(7)∑n=2∞(αn2+(1-α)n-β)|an|r0n-1≤|1-β|,
where r0 is the real root in (0,1) of (6). Using the known expansions
(8)∑n=3∞r0n-1=11-r0-1-r0,(9)∑n=3∞nr0n-1=1(1-r0)2-1-2r0,(10)∑n=3∞n2r0n-1=1+r0(1-r0)3-1-4r0,(11)∑n=3∞n3r0n-1=1+4r0+r02(1-r0)4-1-8r0
leads to
(12)∑n=2∞(αn2+(1-α)n-β)|an|r0n-1≤2(2α+2-β)br0+∑n=3∞(αn2+(1-α)n-β)(cn+d)r0n-1=2(2α+2-β)br0+cα(1+4r0+r02(1-r0)4-1-8r0)+((1-α)c+αd)(1+r0(1-r0)3-1-4r0)+((1-α)d-βc)(1(1-r0)2-1-2r0)-βd(11-r0-1-r0)=(c+d)(β-1)-(2α+2-β)(2(c-b)+d)r0+(cα(1+4r0+r02)+((1-α)c+αd)(1-r02)+((1-α)d-βc)(1-r0)2-βd(1-r0)3)×(1-r0)-4.=|1-β|.
For β<1, consider the function
(13)f0(z)=z-2bz2-∑n=3∞(cn+d)zn=(c+1)z+2(c-b)z2-cz(1-z)2-dz31-z.
At the root z=r0 in (0,1) of (6), f0 satisfies
(14)Re(αz2f′′0(z)f0(z)+zf0′(z)f0(z))=1-N(r0)D(r0)=β,
where
(15)N(r0)=-2(c-b)(2α+1)r0+2cr0(2α+1)(1-r0)3+6cαr02(1-r0)4+2dr02(3α+1)1-r0+dr03(6α+1)(1-r0)2+2dr04α(1-r0)3,D(r0)=c+1+2(c-b)r0-c(1-r0)2-dr021-r0.
This shows that r0 is the sharp L(α,β)-radius for f∈Ab. For β<1, (14) shows that the rational expression N(r0)/D(r0) is positive, and therefore the equality
(16)|αz2f0′′(z)f0(z)+zf0′(z)f0(z)-1|=1-β
holds. Thus, r0 is the sharp L0(α,β)-radius for f∈Ab when β<1.
For β>1, the function
(17)f0(z)=z+2bz2+∑n=3∞(cn+d)zn=(1-c)z+2(b-c)z2+cz(1-z)2+dz31-z
demonstrates sharpness of the result. The derivation is similar to the case β<1 and is omitted.
Theorem 3.
Let β∈R∖{1} and α≥0. The L(α,β)-radius of f(z)=z+∑n=2∞anzn∈Ab satisfying the coefficient inequality |an|≤c/n for n≥3 and c>0 is the real root in (0,1) of the equation
(18)[c(1-β)+|1-β|+(2α+2-β)r(c2-2b)](1-r)2=cα+(1-α)c(1-r)+βc(1-r)2log(1-r)r.
For β<1, this number is also the L0(α,β)-radius of f∈Ab. The results are sharp.
Proof.
By Lemma 1, r0 is the L(α,β)-radius of functions f∈Ab when inequality (7) holds for the real root r0 of (18) in (0,1). Using (8) and (9) together with
(19)∑n=3∞r0n-1n=-log(1-r0)r0-1-r02
leads to
(20)∑n=2∞(αn2+(1-α)n-β)|an|r0n-1≤2(2α+2-β)br0+∑n=3∞(αn2+(1-α)n-β)(cn)r0n-1=2(2α+2-β)br0+cα(1(1-r0)2-1-2r0)+(1-α)c(11-r0-1-r0)-βc(-log(1-r0)r0-1-r02)=c(β-1)+(2α+2-β)r0(2b-c2)+cαr0+(1-α)c(1-r0)r0+βc(1-r0)2log(1-r0)(1-r0)2r0=|1-β|.
To verify sharpness for β<1, consider the function
(21)f0(z)=z-2bz2-∑n=3∞cnzn=(1+c)z+(c2-2b)z2+clog(1-z).
At the root z=r0 in (0,1) of (18), f0 satisfies
(22)Re(αz2f0′′(z)f0(z)+zf0′(z)f0(z))=1-(clog(1-r0)r0-(c2-2b)r0(2α+1)+cr0α(1-r0)2+c1-r0+clog(1-r0)r0-(c2-2b)r0(2α+1)+cr0α(1-r0)2)×((1+c)+(c2-2b)r0+clog(1-r0)r0)-1=β.
Thus, r0 is the sharp L(α,β)-radius for f∈Ab. For β<1, the rational expression in (22) is positive, and therefore
(23)|αz2f0′′(z)f0(z)+zf0′(z)f0(z)-1|=1-β,
which shows that r0 is the sharp L0(α,β)-radius for f∈Ab. For β>1, sharpness of the result is demonstrated by the function f0 given by
(24)f0(z)=z+2bz2+∑n=3∞cnzn=(1-c)z+(2b-c2)z2-clog(1-z).
Remark 4.
The results obtained above yield the following special cases.
For α=0, β=0, c=1, d=0, and 0≤b≤1, Theorem 2 yields the radius of starlikeness obtained by Yamashita [13].
For α=0, c=1, and d=0, Theorem 2 reduces to Theorem 2.1 in [33, page 3]. When α=0, c=0, and d=M, Theorem 2 leads to Theorem 2.5 in [33, page 5].
For α=0, Theorem 3 yields the radius of starlikeness of order β for f∈Ab obtained by Ravichandran [33, Theorem 2.8].
The following result of Goel and Sohi [35] will be required in our investigation of the class of Janowski starlike functions.
Lemma 5 (see [35]).
Let -1≤B<A≤1. If f(z)=z+∑n=2∞anzn∈A satisfies the inequality
(25)∑n=2∞((1-B)n-(1-A))|an|≤A-B,
then f∈ST[A,B].
The next result finds the sharp ST[A,B]-radius for f∈Ab satisfying the coefficient inequality |an|≤cn+d.
Theorem 6.
Let -1≤B<A≤1. The ST[A,B]-radius for f(z)=z+∑n=2∞anzn∈Ab satisfying the coefficient inequality |an|≤cn+d, n≥3 and c,d≥0, is the real root in (0,1) of the equation
(26)[(A-B)(c+d+1)-(2b-2c-d)(2(1-B)-(1-A))r](1-r)3=c(1-B)(1+r)+(d(1-B)-c(1-A))(1-r)-(1-A)d(1-r)2.
This radius is sharp.
Proof.
It is evident that r0 is the ST[A,B]-radius of f∈Ab if and only if f(r0z)/r0∈ST[A,B]. Hence, by Lemma 5, it suffices to show that
(27)∑n=2∞((1-B)n-(1-A))|an|r0n-1≤A-B(-1≤B<A≤1),
where r0 is the root in (0,1) of (26). From (8), (9), and (10), it follows that
(28)∑n=2∞((1-B)n-(1-A))|an|r0n-1≤2(2(1-B)-(1-A))br0+∑n=3∞((1-B)n-(1-A))(cn+d)r0n-1=2(2(1-B)-(1-A))br0+c(1-B)(1+r0(1-r0)3-1-4r0)+(d(1-B)-c(1-A))(1(1-r0)2-1-2r0)-(1-A)d(11-r0-1-r0)=(B-A)(c+d)+(2b-2c-d)×(2(1-B)-(1-A))r0+((1-r0)2c(1-B)(1+r0)+(d(1-B)-c(1-A))(1-r0)-(1-A)d(1-r0)2)×(1-r0)-3=A-B.
The function f0 given by (13) shows that the result is sharp. Indeed, at the point z=r0 where r0 is the root in (0,1) of (26), the function f0 satisfies
(29)|zf0′(z)f0(z)-1|=(-2(c-b)r0+2dr021-r0+dr03(1-r0)2+2cr0(1-r0)3)×(c+1+2(c-b)r0-c(1-r0)2-dr021-r0)-1,|A-Bzf0′(z)f0(z)|=(c+1)(A-B)+2(c-b)r0(A-2B)c+1+2(c-b)r0-c/(1-r0)2-dr02/(1-r0)-((dr03B/(1-r0)2)c(A-B)(1-r0)2+2cr0B(1-r0)3-dr02(A-3B)1-r0+dr03B(1-r0)2)×(c+1+2(c-b)r0-c(1-r0)2-dr021-r0)-1.
Then, (26) yields
(30)|zf0′(z)f0(z)-1|=|A-Bzf0′(z)f0(z)|(-1≤B<A≤1,z=r0),
or equivalently f0∈ST[A,B].
Theorem 7.
Let -1≤B<A≤1. The ST[A,B]-radius for f(z)=z+∑n=2∞anzn∈Ab satisfying the coefficient inequality |an|≤c/n, n≥3 and c>0, is the real root in (0,1) of the equation
(31)((c+1)(A-B)-(2(1-B)-(1-A))r(2b-c2))×(1-r)=c(1-B)+c(1-A)(1-r)log(1-r)r.
This radius is sharp.
Proof.
By Lemma 5, condition (27) assures that r0 is the ST[A,B]-radius of f∈Ab where r0 is the real root of (31). Therefore, using (8) and (19) for f∈Ab yields
(32)∑n=2∞((1-B)n-(1-A))|an|r0n-1≤2(2(1-B)-(1-A))br0+∑n=3∞((1-B)n-(1-A))(cn)r0n-1=2(2(1-B)-(1-A))br0+c(1-B)(11-r0-1-r0)-c(1-A)(-log(1-r0)r0-1-r02)=c(B-A)+(2(1-B)-(1-A))r0(2b-c2)+c(1-B)r0+c(1-A)(1-r0)log(1-r0)(1-r0)r0=A-B.
The result is sharp for the function f0 given by (21). Indeed, f0 satisfies
(33)|zf0′(z)f0(z)-1|=-(c/2-2b)r0+c/(1-r0)+(clog(1-r0))/r0(1+c)+(c/2-2b)r0+(clog(1-r0))/r0,|A-Bzf0′(z)f0(z)|=(cAlog(1-r0)r0(1+c)(A-B)+(A-2B)(c2-2b)r0+cB1-r0+cAlog(1-r0)r0)×((1+c)+(c2-2b)r0+clog(1-r0)r0)-1,
at the root z=r0 in (0,1) of (31). Evidently, the function f0 satisfies (30), and hence the result is sharp.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work presented here was supported in parts by an FRGS Grant 203/PMATHS/6711366 and a grant from the University of Delhi.
de BrangesL.A proof of the Bieberbach conjecture19851541-213715210.1007/BF02392821MR772434ZBLl0573.300142-s2.0-0000441937NevanlinnaR.Uber die konforme Abbildung Sterngebieten1921636121GoodmanA. W.19831Tampa, Fla, USAMariner PublishingMR704183ReadeM. O.On close-to-close univalent functions19553596210.1307/mmj/1031710535MR0070715LowenerC. Untersuchungen uber die Verzerrung bei konformen Abbildungen des Einheitskreises |z|<119176989106SchildA.On a class of univalent, star shaped mappings1958975175710.1090/S0002-9939-1958-0095954-5MR0095954SakaguchiK.On a certain univalent mapping195911727510.2969/jmsj/01110072MR0107005GoodmanA. W.SaffE. B.On the definition of a close-to-convex function19781112513210.1155/S0161171278000150MR0480971ZBLl0383.30005GoodmanA. W.On uniformly starlike functions1991155236437010.1016/0022-247X(91)90006-LMR1097287ZBLl0726.300132-s2.0-0001561856GoodmanA. W.On uniformly convex functions19915618792MR1145573GavrilovV. I.Remarks on the radius of univalence of holomorphic functions19707295298MR0260989LandauE.Der Picard-Schottkysche Satz und die Blochsche Konstante1925467474YamashitaS.Radii of univalence, starlikeness, and convexity198225345345710.1017/S0004972700005499MR671491ZBLl0481.30012KalajD.PonnusamyS.VuorinenM.Radius of close-to-convexity and fully starlikeness of harmonic mappings201459453955210.1080/17476933.2012.759565MR 77008LiuZ. W.LiuM. S.Properties and characteristics of certain subclass of analytic functions201031114, 18LewandowskiZ.MillerS.ZlotkiewiczE.Generating functions for some classes of univalent functions19765611111710.1090/S0002-9939-1976-0399438-7MR0399438RameshaC.KumarS.PadmanabhanK. S.A sufficient condition for starlikeness1995232167171MR1340763ZBLl0834.30010NunokawaM.OwaS.LeeS. K.ObradovicM.AoufM. K.SaitohH.IkedaA.KoikeN.Sufficient conditions for starlikeness1996243265271MR1408788ObradovicM.JoshiS. B.On certain classes of strongly starlike functions199823297302MR16411592-s2.0-0006737255LiJ.OwaS.Sufficient conditions for starlikeness2002333313318MR1894627ZBLl0998.300102-s2.0-0036015273PadmanabhanK. S.On sufficient conditions for starlikeness2001324543550MR1838830ZBLl0979.300072-s2.0-0035535961RavichandranV.SelvarajC.RajalaksmiR.Sufficient conditions for starlike functions of order α200235, article 816MR1966516RavichandranV.Certain applications of first order differential subordination20041214151MR2026702ZBLl1061.30014LiuM. S.ZhuY. C.SrivastavaH. M.Properties and characteristics of certain subclasses of starlike functions of order β2008483-440241910.1016/j.mcm.2006.09.026MR24314752-s2.0-44849140805NishiwakiJ.OwaS.Coefficient inequalities for certain analytic functions200229528529010.1155/S0161171202006890MR1896244ZBLl1003.300062-s2.0-17844410637UralegaddiB. A.GanigiM. D.SarangiS. M.Univalent functions with positive coefficients1994253225230MR1304483ZBLl0837.30012UralegaddiB. A.DesaiA. R.Convolutions of univalent functions with positive coefficients1998294279285MR1648530ZBLl0926.30007OwaS.SrivastavaH. M.Some generalized convolution properties associated with certain subclasses of analytic functions200233, article 4213MR1917801JanowskiW.Some extremal problems for certain families of analytic functions—I197328297326MR0328059AliR. M.ChoN. E.JainN. K. a.Radii of starlikeness and convexity for functions with fixed second coefficient defined by subordination201226355356110.2298/FIL1203553AMR30982602-s2.0-84867722810AliR. M.NagpalS.RavichandranV.Second-order differential subordination for analytic functions with fixed initial coefficient2011343611629MR28235922-s2.0-80051485078NagpalS.RavichandranV.Applications of the theory of differential subordination for functions with fixed initial coefficient to univalent functions2012105322523810.4064/ap105-3-2MR2950658ZBLl1267.300492-s2.0-84864393728RavichandranV.Radii of starlikeness and convexity of analytic functions satisfying certain coefficient inequalities2014641273810.2478/s12175-013-0184-4MR3174256SunY.WangZ.XiaoR.Neighbourhoods and partial sums of certain subclass of analytic functions201126217224MR2850613GoelR. M.SohiN. S.Multivalent functions with negative coefficients1981127844853MR624534ZBLl0463.30016