The main purpose of this paper is to introduce and investigate a certain subclass of meromorphic close-to-convex functions. Such results as coefficient inequalities, convolution property, inclusion relationship, distortion property, and radius of meromorphic convexity are derived.
1. Introduction and Preliminaries
Let Σ denote the class of functions f of the form
(1)f(z)=1z+∑n=1∞anzn,
which are analytic in the punctured open unit disk
(2)U*∶={z:z∈C0<|z|<1}=:U∖{0}.
For two functions f and g, analytic in U, we say that the function f is subordinate to g in U and write
(3)f(z)≺g(z)(z∈U),
if there exists a Schwarz function ω, which is analytic in U with
(4)ω(0)=0,|ω(z)|<1(z∈U),
such that
(5)f(z)=g(ω(z))(z∈U).
Indeed, it is known that
(6)f(z)≺g(z)(z∈U)⟹f(0)=g(0),11111111111111111111f(U)⊂g(U).
Furthermore, if the function g is univalent in U, then we have the following equivalence:
(7)f(z)≺g(z)(z∈U)⟺f(0)=g(0),11111111111111111111f(U)⊂g(U).
Let f,g∈Σ, where f is given by (1) and g is defined by
(8)g(z)=1z+∑n=1∞bnzn.
Then the Hadamard product (or convolution) f*g of the functions f and g is defined by
(9)(f*g)(z)∶=1z+∑n=1∞anbnzn=:(g*f)(z).
A function f∈Σ is said to be in the class MS* of meromorphic starlike functions if it satisfies the inequality
(10)R(zf′(z)f(z))<0(z∈U).
A function f∈Σ is said to be in the class MK of meromorphic convex functions if it satisfies the inequality
(11)R(1+zf′′(z)f′(z))<0(z∈U).
Moreover, a function f∈Σ is said to be in the class MC of meromorphic close-to-convex functions if it satisfies the condition
(12)R(zf′(z)g(z))<0(z∈U;g∈MS*).
Let
(13)f(z)=z+a2z2+a3z3+⋯
be analytic in U. If there exists a function g∈K such that
(14)|(zf′(z)/g(z))-1(zf′(z)/g(z))+(1-2α)|<β(z∈U;0≦α<1;0<β≦1),
then we say that f∈C(α,β), where K denotes the usual class of convex functions. The function class C(α,β) was introduced and studied recently by Peng [1] (see also Peng and Han [2], Selvaraj [3], Gao and Zhou [4], Kowalczyk and Leś-Bomba [5], and Xu et al. [6]).
Motivated essentially by the above mentioned function class C(α,β), we now introduce and investigate the following class of meromorphic close-to-convex functions.
Definition 1.
A function f∈Σ is said to be in the class MC(α,β) if it satisfies the inequality
(15)|(zf′(z)/g(z))+1(zf′(z)/g(z))+(2α-1)|<β(z∈U;g∈MS*),
where (and throughout this paper unless otherwise mentioned) the parameters α and β are constrained as follows:
(16)0≦α<1,0<β≦1.
It is easy to verify that f∈MC(α,β) if and only if
(17)-zf′(z)g(z)≺1+(1-2α)βz1-βz(z∈U).
We observe that
(18)R(1+(1-2α)βz1-βz)>0(z∈U),
and, thus, the function class MC(α,β) is a subclass of meromorphic close-to-convex functions.
Clearly, the class MC(α,1)=:MC(α) is the familiar class of meromorphic close-to-convex functions of order α.
For some recent investigations of meromorphic functions, see, for example, the works of [7–22] and the references cited therein.
To derive our main results, we need the following lemmas.
Lemma 2 (see [23]).
Let
(19)f1(z)=1+∑n=1∞cnzn
be analytic in U and let
(20)f2(z)=1+∑n=1∞dnzn
be analytic and convex in U. If f1≺f2, then
(21)|cn|≦|d1|(n∈N≔{1,2,…}).
Lemma 3 (see [24]).
Suppose that
(22)h(z)=1z+∑n=1∞cnzn∈MS*.
Then
(23)|cn|≦2n+1(n∈N).
Each of these inequalities is sharp, with the extremal function given by
(24)h(z)=z-1(1+zn+1)2/(n+1).
Lemma 4 (see [25]).
Let -1≦D<C≦1 and -1≦F<E≦1. Then,
(25)1+Cz1+Dz≺1+Ez1+Fz
if and only if
(26)|ED-FC|≦(E-F)-(C-D).
Lemma 5 (see [26]).
Suppose that the function g∈MS*. Then
(27)(1-r)2r≦|g(z)|≦(1+r)2r(|z|=r;0<r<1).
Lemma 6 (see [27]).
Suppose that
(28)p(z)≺1+(1-2α)z1-z(z∈U;0≦α<1).
Then,(29)R(zp′(z)p(z))≧{-2(1-α)r(1+r)[1+(2α-1)r](|z|=r;0<r≦r0),(4α(1-r2)[1+(1-2α)r2]-[1+(1-2α)r2]4α(1-r2)[1+(1-2α)r2])×((1-α)(1-r2))-1-α1-α(|z|=r;r0<r<1),
where r0 is the unique root of the equation
(30)(2α-1)r4-2(2α-1)r3-6αr2-2r+1=0
in the interval (0,1). The results are sharp.
In the present paper, we aim at proving some coefficient inequalities, convolution property, inclusion relationship, distortion property, and radius of meromorphic convexity of the class MC(α,β).
2. Main Results
We begin by stating the following coefficient inequality of the class MC(α,β).
Theorem 7.
Suppose that
(31)f(z)=1z+∑n=1∞anzn∈MC(α,β).
Then
(32)|a1|≦1,(33)|an|≦2(1-α)βn(1+∑k=1n-12k+1)+2n(n+1)111111111111111111111111111(n∈N∖{1}).
Proof.
Let f∈MC(α,β) and suppose that
(34)p(z)∶=-zf′(z)g(z)(z∈U),
where
(35)g(z)=1z+∑n=1∞bnzn∈MS*.
It follows that
(36)p(z)=1+∑n=1∞pnzn≺1+(1-2α)βz1-βz(z∈U).
In view of Lemma 2, we know that
(37)|pn|≦2(1-α)β(n∈N).
By substituting the series expressions of functions f, g, and p into (34), we get
(38)(1+p1z+p2z2+⋯+pnzn+pn+1zn+1+⋯)×(1z+b1z+b2z2+⋯+bnzn+⋯)=1z-a1z-2a2z2-⋯-nanzn-⋯.
Since f is univalent in U*, it is well known that |a1|≦1.
On the other hand, we find from (38) that
(39)-nan=bn+p1bn-1+p2bn-2+⋯+pn-1b1+pn+1111111111111111111111111111111(n∈N∖{1}).
By noting that g∈MS*, it follows from Lemma 3 that
(40)|bn|≦2n+1(n∈N).
Combining (37), (39), and (40), we have
(41)n|an|≦2(1-α)β(2n+2n-1+⋯+23+1+1)+2n+111111111111111111111111111111111111(n∈N∖{1}).
Thus, the assertion (33) of Theorem 7 follows directly from (41).
Theorem 8.
Let
(42)g(z)=1z+∑n=1∞bnzn∈MS*.
If f∈Σ satisfies the condition
(43)(1+β)∑n=1∞n|an|+(1+|2α-1|β)∑n=1∞|bn|≦2(1-α)β,
then f∈MC(α,β).
Proof.
To prove f∈MC(α,β), it suffices to show that (15) holds. From (43), we know that
(44)β(2(1-α)-∑n=1∞n|an|-|2α-1|∑n=1∞|bn|)≧∑n=1∞n|an|+∑n=1∞|bn|>0.
Now, by the maximum modulus principle, we deduce from (1) and (44) that
(45)|(zf′(z)/g(z))+1(zf′(z)/g(z))+(2α-1)|=|∑n=1∞nanzn+1+∑n=1∞bnzn+1∑n=1∞nanzn+1+(2α-1)∑n=1∞bnzn+1+2(α-1)|<∑n=1∞n|an|+∑n=1∞|bn|2(1-α)-∑n=1∞n|an|-|2α-1|∑n=1∞|bn|≦β.
This evidently completes the proof of Theorem 8.
Example 9.
By applying Theorem 8, it is obvious to see that the function
(46)f(z)=1z+z∈MC(0,1).
Theorem 10.
Let |ξ|=1 and g∈MS*. A function f∈MC(α,β) if and only if
(47)f(z)*[(1-βξ)1-2zz(1-z)2]-g(z)*{[1+(1-2α)βξ]z2-z+1z(1-z)}≠0111111111111111111111111(z∈U*).
Proof.
A function f∈MC(α,β) if and only if
(48)-zf′(z)g(z)≠1+(1-2α)βξ1-βξ(z∈U;|ξ|=1).
It is easy to see that condition (48) can be written as
(49)zf′(z)(1-βξ)+g(z)[1+(1-2α)βξ]≠011111111111111111111111(z∈U*;|ξ|=1).
We observe that
(50)-zf′(z)=f(z)*(1z-z(1-z)2)=f(z)*1-2zz(1-z)2,g(z)=g(z)*(1z+z1-z)=g(z)*z2-z+1z(1-z).
By substituting (50) into (49), we get the desired assertion (47) of Theorem 10.
Theorem 11.
Let
(51)0≦α2≦α1≦12,0<β1≦β2<1.
Then,
(52)MC(α1,β1)⊂MC(α2,β2).
Proof.
Suppose that f∈MC(α1,β1). We easily know that
(53)-zf′(z)g(z)≺1+(1-2α1)β1z1-β1z(z∈U).
By setting C=1+(1-2α1)β1, D=-β1, E=1+(1-2α2)β2, and F=-β2, it follows from (51) that
(54)|ED-FC|=|-(1-2α2)β1β2+(1-2α1)β1β2|≦|-(1-2α2)β1β2+(1-2α1)β12|111+|-(1-2α1)β12+(1-2α1)β1β2|≦[(1-2α2)β2-(1-2α1)β1]+(β2-β1)=[(1-2α2)β2-(-β2)]111-[(1-2α1)β1-(-β1)]=(E-F)-(C-D).
In view of Lemma 4, we deduce that
(55)-zf′(z)g(z)≺1+(1-2α1)β1z1-β1z≺1+(1-2α2)β2z1-β2z11111111111111111111111111111111111(z∈U),
which implies that f∈MC(α2,β2). Thus, the assertion (52) of Theorem 11 holds.
Theorem 12.
Let f∈MC(α,β). Then,
(56)(1-r)2[1-(1-2α)βr]r2(1+βr)≦|f′(z)|≦(1+r)2[1+(1-2α)βr]r2(1-βr)1111111111111(|z|=r;0<r<1).
Proof.
Let f∈MC(α,β). By definition, we know that
(57)-zf′(z)g(z)≺1+(1-2α)βz1-βz(z∈U).
Suppose that the function p is defined by (36). Then, we have
(58)1-(1-2α)βr1+βr≦|p(z)|≦1+(1-2α)βr1-βr,11111111111111111(|z|=r;0<r<1).
Since g∈MS*, by Lemma 5, we know that
(59)(1-r)2r≦|g(z)|≦(1+r)2r(|z|=r;0<r<1).
Thus, by virtue of (36), (58), and (59), we readily get the assertion (56) of Theorem 12.
Finally, we derive the radius of meromorphic convexity for the class MC(α).
Theorem 13.
Let f∈MC(α) with 0<α<1. Then,
for r1≦r0, f is meromorphic convex in 0<|z|<r1;
for r1>r0, f is meromorphic convex in 0<|z|<r2,
where r0 is the unique root of the equation
(60)(2α-1)r4-2(2α-1)r3-6αr2-2r+1=0
in the interval (0,1) and r1 and r2 are the smallest root of the equations
(61)(2α-1)r3+3(2α-1)r2+3r+1=0,(1-2α)r4+2(1-2α)r3+3(1-α)r2+2r+1=0
in the interval (0,1), respectively.
Proof.
Let f∈MC(α) and suppose that
(62)q(z)∶=-zf′(z)g(z)(z∈U).
Then,
(63)q(z)≺1+(1-2α)z1-z(z∈U).
It follows from (62) that
(64)-zf′(z)=q(z)g(z).
Differentiating both sides of (64) logarithmically, we get
(65)-(1+zf′′(z)f′(z))=zq′(z)q(z)+zg′(z)g(z).
Since g∈MS*, we know that
(66)R(zg′(z)g(z))≧-1+r1-r(|z|=r).
Combining (63), (65), (66), and Lemma 6, we obtain
(67)-R(1+zf′′(z)f′(z))≧{-1+r1-r-2(1-α)r(1+r)[1+(2α-1)r]=:H~α(r)11111111111111111(|z|=r;0<r≦r0),-1+r1-r+(4α(1-r2)[1+(1-2α)r2]-[1+(1-2α)r2]4α(1-r2)[1+(1-2α)r2])×((1-α)(1-r2))-1-α1-α=:F~α(r)1111111111(|z|=r;r0<r<1),
where r0 is the unique root of (30) in the interval (0,1). It follows from (67) that the bound of meromorphic convexity for the class MC(α) is determined either by the equation
(68)-1+r1-r-2(1-α)r(1+r)[1+(2α-1)r]=0,
or by the equation
(69)-1+r1-r+4α(1-r2)[1+(1-2α)r2]-[1+(1-2α)r2](1-α)(1-r2)-α1-α=0.
We note that (68) and (69) can be rewritten as follows:(70)Hα(r)∶=(2α-1)r3+3(2α-1)r2+3r+1=0,Fα(r)∶=(1-2α)r4+2(1-2α)r3+3(1-α)r2+2r+1=0.
Let r1 and r2 be the smallest root of the equations Hα(r)=0 and Fα(r)=0 in the interval (0,1), respectively. By observing that Hα(0)=1>0, we deduce that Hα(r)>0 for r<r1.
Similarly, we know that Fα(r)>0 for r<r2, since Fα(0)=1>0.
We observe that
(71)H~α(r)≧0⟺Hα(r)≧0,F~α(r)≧0⟺Fα(r)≧0.
Thus, when r1≦r0, f is meromorphic convex in 0<|z|<r1; when r1>r0, f is meromorphic convex in 0<|z|<r2.
The proof of Theorem 13 is thus completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation under Grant nos. 11301008 and 11226088, the Foundation for Excellent Youth Teachers of Colleges and Universities of Henan Province under Grant no. 2013GGJS-146, and the Natural Science Foundation of Educational Committee of Henan Province under Grant no. 14B110012 of the People's Republic of China. The authors would like to thank the referees for their valuable comments and suggestions which essentially improved the quality of this paper.
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