# On Some Problems of Strongly Ozaki Close-to-Convex Functions

Authors: Zahra Maleki; Saeid Shams; Ali Ebadian; Ebrahim Analouei Adegani
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The purpose of the current paper is to investigate some geometric properties of the classFOν,γ, called strongly Ozaki close-to-convex functions, such as strongly starlikeness and close-to-convexity. Further, we find sharp bounds on Fekete-Szegö functionals and logarithmic coefficients for functions belonging to the class FOν,γ, which incorporates some known outcomes as the specific cases.


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## Body

## 1. Introduction and Preliminaries

LetU denote the open unit dick in the complex plane ℂ. Let A be the class of functions f of the following normalized form:
(1)fz=z+∑n=2∞anzn,which are analytic in U and represent by S the class of all functions of A, which are univalent in U. Let Ω denote the set of all analytic functions ω in U that are satisfying the conditions of ω0=0 and ωz<1 for z∈U, i.e., Ω, is considered as the family of Schwarz functions.For two analytic functionsf and F in the open unit dick U, it is said that the function f is subordinate to the function F in U, written fz≺Fz, if there exists a Schwarz function ω such that fz=Fωz for all z∈U. In particular, if the function F is univalent in U, the following equivalence holds:
(2)fz≺Fz⇔f0=F0,fU⊂FU.We denote byS∗α the subclass of A consisting of all f∈A for which f is a starlike of order α, with
(3)Rezf′zfz>αz∈U,0≤α<1,and denote by Kα the subclass of A consisting of all f∈A for which f is a convex of order α, with
(4)1+Rezf″zf′z>αz∈U,0≤α<1.Also, the subclassS~∗α of a strongly starlike function of order in U is defined as
(5)S~∗α=f∈A:Argzf′zfz<απ2,z∈U,0≤α<1.Note thatS~∗1=S∗0=S∗ and K0=K are the class of starlike functions in U and the class of convex functions in U, respectively.Furthermore,Cα is denoted as the subclass of A including functions such as close-to-convex of order α if there is a function g∈S∗ so that
(6)Rezf′zgz>αz∈U,0≤α<1,and we denote by C~α the subclass of A consisting of all of f∈A for which
(7)Argf′z<απ2z∈U,0≤α<1.Individually,C0=C is the class of close-to-convex functions in U and C~1=C~ is the subclass of close-to-convex functions in U (see [1]). Here, we understand that Argw is a number in −π,π.Recently, many authors have studied the families of analytic functions of the classA and also investigated bound estimation problems, geometric property issues, and related topics for functions belonging to these families in [2–10] as well as in the references cited therein.For example, Cho et al. [4] studied the majorization issue for a general well-known category S∗φ of starlike functions, which was defined by Ma and Minda [11]. Also, they investigated the majorization issue for the various subclasses S∗φ for different special functions φ. Moreover, estimates for the coefficients of majorized functions regarding the class S∗φ were given. Further, Alimohammadi et al. [2] introduced a subclass of A and extended the class Gαα∈0,1, defined by Nunokawa et al. in [12], consisting of all f∈A satisfying
(8)Re1+zf″zf′z<1+α2z∈U,and studied some geometric properties like close-to-convexity and strongly starlikeness. They determined sharp bounds of Fekete-Szegö functionals and logarithmic coefficients for this class. Kargar and Ebadian [9] considered the subclass Fν of locally univalent functions f∈A in U satisfying the inequality:
(9)Re1+zf″zf′z>1−ν2z∈U,for some −1/2<ν≤1.Recently, Allu et al. [3], motivated essentially by the subclass Fν, introduced the class FOν,γ and obtained sharp bounds for three first coefficients and the corresponding inverse coefficients for the functions of this class.Definition 1 [3].
Letγ∈0,1 and ν∈1/2,1. Then, f is called strongly Ozaki close-to-convex if and only if
(10)FOν,γ≔f∈A:Arg2ν−12ν+1+22ν+11+zf″zf′z<γπ2,z∈U.Note that the classFOν,1=FOν was introduced in [9] and members of this class were called Ozaki close-to-convex functions. Also, FO1,1=F1 was studied by Ponnusamy et al. [13]. Furthermore, FO1/2,1=K.It is remarkable that by means of the principle of subordination between analytic functions, the definition of the classFOν,γ can be rewritten as follows:
(11)FOν,γ≔f∈A:1+zf″zf′z≺2ν+121+z1−zγ−2ν−12≔ϕz,z∈U.The present paper was undertaken to investigate some geometric features of the classFOν,γ such as close-to-convexity and strongly starlikeness. In addition, we found estimates for the coefficients an and give sharp bounds on Fekete-Szegö functionals and logarithmic coefficients for functions belonging to the class FOν,γ, which incorporates some known outcomes as the specific cases.


## 2. Some Geometric Properties of the ClassFOν,γ

In this section, we investigate some geometric properties like strongly starlikeness and close-to-convexity for the classFOν,γ to present the relation of this class with the well-known families of univalent functions. The key in proving is Nunokawa’s lemma [14] (see also [15]), and so in order to prove our result, we require the following lemmas.We denote byQ the class of all complex-valued functions q for which q is univalent at each U¯\Eq and q′ξ≠0 for all ξ∈∂U\Eq where
(12)Eq=ξ∈∂U:limz→ξqz=∞.Lemma 2 ([16], Lemma 2.2d (i)).
Letq∈Q with q0=a and let pz=a+pnzn+⋯ be analytic in U with pz≡1 and n≥1. If p is not subordinate to q in U, then there exist z0∈U and ξ0∈∂U\Eq such that pz:z∈U,z<z0⊂qU:
(13)pz0=qξ0.Lemma 3 (see [14, 15]).
Let the functionp given by
(14)pz=1+∑n=1∞cnznbe analytic in U with p0=1 and pz≠0 for all z∈U. If there exists a point z0∈U with
(15)argpz<βπ2z<z0,argpz0=βπ2,for some β>0, then
(16)z0p′z0pz0=ikβi=−1,where
(17)k≥a+a−12≥1,whenargpz0=βπ2(18)k≤−a+a−12≤−1,whenargpz0=−βπ2,where
(19)pz01/β=±ia,a>0.Theorem 4.
Letν∈1/2,1 and β0≤β≤1 where β00<β0<1 is given by tanπ/2β0=2/2ν−1β0. If f∈A satisfies the following condition:
(20)Arg2ν−12ν+1+22ν+11+zf″zf′z<Arctan2β2ν−1,then
(21)Argzf′zfz<βπ2z∈U.Proof.
The result is proven by contradiction. Letf∈A and define the function M:U⟶ℂ by
(22)Mz=zf′zfz=1+∑n=1∞cnznz∈U.
Then, it is concluded thatM is analytic in U, M0=1,
(23)1+zf″zf′z=Mz+zM′zMzz∈U,and Mz≠0 for all z∈U. Indeed, if M has a zero z0∈U of order m, then we have
(24)Mz=z−z0mM1zm∈ℕ=1,2,3,⋯,where M1 is analytic in U with M1z0≠0. Then,
(25)2ν−12ν+1+22ν+1Mz+zM′zMz=2ν−12ν+1+22ν+1Mz+zM1′zM1z+mzz−z0.
Hence, withz⟶z0, in the right hand of the above equality, the argument can properly take any value between −π and π, which contradicts to (20).
Define the functionN:U¯\1⟶ℂ by
(26)Nz=1+z1−zβz∈U¯\1.
Then,N∈Q, N0=1, and EN=1. Clearly, ArgMz<βπ/2z∈U if and only if M≺N on U. Suppose ArgMz1≥βπ/2 for some z1∈U. Then, M is not subordinate to N. By applying Lemma 2, there exist z0∈U and ξ0∈∂U\1 so that Mz:z∈U,z<z0⊂NU and Mz0=Nξ0. Thus,
(27)ArgMz<βπ2z∈U,with z<z0 and
(28)ArgMz0=βπ2.
Therefore, Lemma3 results in
(29)zM′z0Mz0=ikβ,where Mz¯01/β=±iaa>0 and κ is stated by (17) or (18).
First, letArgMz0=βπ/2. Then, we write Mz0=aβcosβπ/2+isinβπ/2, and so for k≥1, we have
(30)Arg2ν−12ν+11+22ν−1Mz0+z0M′z0Mz0=Arg1+22ν−1aβcosβπ2+i22ν−1aβsinβπ2+kβ=Arctan2/2ν−1aβsinβπ/2+kβ1+2/2ν−1aβcosβπ/2≥Arctan2/2ν−1aβsinβπ/2+β1+2/2ν−1aβcosβπ/2.
We define the functionh:0,a⟶ℝ by
(31)hs=2/2ν−1sβsinβπ/2+β1+2/2ν−1sβcosβπ/2,s∈0,a.
Then,h is a differentiable function on 0,a, and
(32)h′s=2β/2ν−1sβcosβπ/2tanβπ/2−2β/2ν−11+2/2ν−1sβcosβπ/22,s∈0,a.
Now, we definegβ as
(33)gβ=tanβπ2−2β2ν−10<β0≤β≤1;β0≠1.
Then,g0=gβ0=0, g′0=π/2−2/2ν−1<0 for ν∈1/2,1, and
(34)g″β=π2sec2βπ2tanβπ2>0.
So, the functiong has a negative value on 0,β0 and a positive value on β0,1.
According to the assumption and from (32), it follows that h′s>0 for all s∈0,a for β∈β0,1. Also, this shows that the function l:0,a⟶ℝ defined by
(35)ls=Arctanhs,s∈0,a,is nondecreasing on 0,a. Hence,
(36)la≥lims→0+lt=Arctan2β2ν−1.
Therefore, we get(37)Arctan2/2ν−1aβsinβπ/2+β1+2/2ν−1aβcosβπ/2≥Arctan2β2ν−1.
Now applying (30) and (37), we obtain
(38)Arg2+α2ν−11+22ν−1pz0+z0p′z0pz0=Arg1+22ν−1pz0+z0p′z0pz0=Arg1+22ν−11+z0f″z0f′z0≥Arctan2/2ν−1aβsinβπ/2+β1+2/2ν−1aβcosβπ/2≥Arctan2β2ν−1,which contradicts to (20).
Next, letArgMz0=−βπ/2. Then, we write Mz0=aβcosβπ/2−isinβπ/2. Thus, for k≤−1 and utilizing (37), we get
(39)Arg2ν−12ν+11+22ν−1Mz0+z0M′z0Mz0=Arg1+22ν−1aβe−iβπ/2+ikβ=Arctan2/2ν−1−aβsinβπ/2+kβ1+2/2ν−1aβcosβπ/2≤Arctan2/2ν−1−aβsinβπ/2−β1+2/2ν−1aβcosβπ/2=−Arctan2/2ν−1aβsinβπ/2+β1+2/2ν−1aβcosβπ/2≤−Arctan2β2ν−1,which contradicts to (20).
From the above contradictions, it results in(40)Argzf′zfz<βπ2z∈U.
Hence, theproof is completed.Corollary 5.
Letν∈1/2,1,β0≤β≤1 where β00<β0<1 is given by tanπ/2β0=2/2ν−1β0 and γ=2/πArctan2β/2ν−1. If f∈FOν,γ, then f∈S~∗β.Remark 6.
According to the assumptions of Corollary5, if f∈FOν,γ, then f∈S~∗β implies Rezf′z/fz>0z∈U and so it is well known that f is univalent.Theorem 7.
Letβ∈0,1,ν∈1/2,1. If f∈A and
(41)Arg2ν+12ν−1+22ν−11+zf″zf′z<Arctan2β2ν+1,then
(42)Argf′z<βπ2z∈U.Proof.
The result is proven by contradiction. To prove our result, we set the functionM:U⟶ℂ by
(43)Mz=f′z=1+∑n=1∞cnznz∈U.
Then,M is analytic in U, p0=1, Mz≠0 for all z∈U, and
(44)1+zf″zf′z=1+zM′zMz.
If there is a pointz0∈U, then
(45)ArgMz<βπ2z∈U,with z<z0, and
(46)ArgMz0=βπ2.
Then, from Lemma3, we have
(47)zM′z0Mz0=ikβ,where Mz01/β=±iaa>0 and k is stated by (17) or (18).
For the caseArgMz0=απ/2when
(48)Mz01/β=iaa>0and k≥1, we have
(49)Arg2ν−12ν+11+22ν−11+z0M′z0Mz0=Arg1+22ν−11+z0M′z0Mz0=Arg1+22ν−11+ikβ=Arctan2kβ2ν+1≥Arctan2β2ν+1,which contradicts to (41).
Next, for the caseArgMz0=−απ/2 when
(50)Mz0=−iaa>0and k≤−1, by applying the same method mentioned above, it can be concluded that
(51)Arg2ν−12ν+11+22ν−11+z0M′z0Mz0=Arg1+22ν−11+z0p′z0pz0=Arg1+22ν−11+ikβ=Arctan−2kβ2ν+1≤−Arctan2β2ν+1,which contradicts to (41).
As a result, from the above contradictions, we obtain(52)Argf′z<βπ2z∈U,and therefore, the proof is completed.Corollary 8.
Letβ∈0,1, ν∈1/2,1, and γ=2/πArctan2β/2ν+1. If f∈FOν,γ, then f∈C~β.Remark 9.
>According to the assumptions of Corollary8, if f∈FOν,γ, then f∈C~β implies Ref′z>0z∈U and so it is well known that f is univalent.


## 3. Coefficient Bounds

In this section, we find sharp bounds on Fekete-Szegö functionals and logarithmic coefficients (see [17–22]) for functions belonging to the class FOν,γ. Also, we present a general problem of coefficients in this class. To prove our main results, some requirements are needed. We remark in passing that the logarithmic coefficients γn of f∈S are defined by the next form:
(53)logfzz=2∑n=1∞γnfznz∈U.These coefficients are of great significance for different estimates in the theory of univalent functions (see [19–21]).Ma and Minda [11] defined the class consisting of several well-known classes as follows:
(54)Kφ≔f∈A:1+zf″zf′z≺φz,z∈U,where in here, it is supposed that φ is a univalent function in U with φ0=1 such that it has the following form:
(55)φz=1+B1z+B2z2+B3z3+⋯,z∈U,withB1≠0.Lemma 10 ([17], Theorem 2).
Letf∈Kφ. Then, the logarithmic coefficients of f satisfy
(56)γ1≤B14,(57)γ2≤B112,if4B2+B12≤4B1,4B2+B128,if4B2+B12>4B1,and if B1, B2, and B3 are real values,
(58)γ3≤B124Hq1;q2,where Hq1;q2 is given by [23, 24], q1=B1+4B2/B1/2, and q2=B2+2B3/B1/2. The bounds (56) and (57) are sharp.Lemma 11 [23, 24].
Ifω∈Ω with ωz=∑n=1∞wnznz∈U, then, the following sharp estimate is given:
(59)w3+q1w1w2+q2w13≤Lq1;q2,for any real numbers q1 and q2 where
(60)Lq1;q2=1,ifq1,q2∈A∪B,q2,ifq1,q2∈C∪D,and the sets A,B,C,D are stated as follows:
(61)A=q1,q2∈ℝ2:q1≤12,q2≤1,B=q1,q2:12≤q1≤2,427q1+13−q1+1≤q2≤1,C=q1,q2:q1≤2,q2≥1,D=q1,q2:2≤q1≤4,q2≥112q12+8.Lemma 12 [23, 25].
Letω∈Ω with ωz=∑n=1∞wnzn for all z∈U. Then,
(62)w2−tw12≤max1,t,forallt∈ℂ.
The result is sharp for the functionωz=z2 or ωz=z.Lemma 13 [26].
Letφ be a convex function in U with the form φz=1+∑n=1∞Bnzn. If function f∈Kφ, then
(63)an≤∏k=2nk−2+B1n!n=2,3,⋯.Theorem 14.
Letf∈FOν,γ. Then,
(64)γ1≤2ν+1γ4,γ2≤2ν+1γ12,ifγ∈0,45+2v,γ22ν+12ν+548,ifγ∈45+2v,1,γ3≤2ν+1γ24,ifγ∈0,26v+7,γ22ν+1γ26v+7+2144,ifγ∈26v+7,1.
All the inequalities are sharp.Proof.
Letf∈FOν,γ. From (11), it follows that
(65)1+zf″zf′z≺ϕz=1+2ν+1γz+2ν+1γ2z2+132ν+1γ2γ2+1z3+⋯=1+B1z+B2z2+B3z3+⋯.
Now, it is enough to obtain the estimate ofγ3. By Lemma 10 with φ=ϕ for γ3, we obtain
(66)γ3≤B124Hq1;q2=2ν+1γ24Hq1;q2=2ν+1γ24Lq1;q2,where Lq1;q2 is given by Lemma 11,
(67)q1=B1+4B2/B12≕γ2ν+52,q2=B2+2B3/B12≕γ26ν+7+26.
First, if we consider(68)A=q1,q2∈ℝ2:q1≤12,q2≤1,then it is clear that q1≤1/2 for 0<γ≤1/2ν+5 and q2≤1 for 0<γ≤2/√6ν+7. Therefore, we conclude q1,q2∈A for 0<γ≤1/2ν+5.
Also, regarding(69)B=q1,q2∈ℝ2:12≤q1≤2,427q1+13−q1+1≤q2≤1,then it is clear that 1/2≤q1≤2 for 1/2ν+5≤γ≤4/2ν+5. Thus, in the above relation, q2≤1 is equivalent to 0<γ≤2/√6ν+7. Also,
(70)427q1+13−q1+1≤q2,that is,
(71)gν,γ≔4ν+5/2327γ3+4ν+5/2227+ν+528ν27+2027−ν−76γ2+−59ν−2518γ−3227≤0holds for 0<γ≤2/√6ν+7 (see Figure 1). Therefore, considering the above results, we conclude q1,q2∈B for γ∈1/2ν+5,2/√6ν+7.Figure 1
The plot ofgν,γ for γ=0⋯2/√6ν+7,ν=1/2⋯1.Next, considering(72)C=q1,q2∈ℝ2:q1≤2,q2≥1,it is clear that q1≤2 for γ≤4/2ν+5, and according to the above computation, the inequality q2≥1 holds for γ≥2/√6ν+7. Therefore, q1,q2∈C for γ∈2/√6ν+7,4/2ν+5.Finally, let us consider(73)D=q1,q2∈ℝ2:2≤q1≤4,q2≥112q12+8.It is clear that2≤q1≤4 for γ∈4/2ν+5,8/2ν+5. On the other hand, the inequality q2≥q12+8/12 is equivalent to
(74)γ2−16−4ν2+28ν+31≥0,and this last inequality holds for γ≥4/√−4ν2+28ν+31. Therefore, q1,q2∈D for γ∈4/√−4ν2+28ν+31,8/2ν+5. Since 8/2ν+5≥1, so q1,q2∈D for γ∈4/√−4ν2+28ν+31,1.By applying the above four conclusions from (66), it follows that
(75)γ3≤2ν+1γ24Lq1;q2=2ν+1γ24,ifγ∈0,26v+7seeAandB,2ν+1γ24q2,ifγ∈26v+7,1seeAandB,=2ν+1γ24,ifγ∈0,26v+7,γ2ν+1γ26v+7+2144,ifγ∈26v+7,1.Suppose thatfν,γ,n∈FOν,γ such that
(76)1+zf″zf′z=ϕznz∈U;n∈ℕ.Then, the functions fν,γ,n are of the following form:
(77)fν,γ,nz=∫0zexp∫0xϕtn−1tdtdx=z+γ2ν+1nn+1zn+1+γ22ν+12ν+n+12n22n+1z2n+1+⋯z∈U;n∈ℕ,and the fν,γ,n are extremal functions for diverse problems in the subclass FOν,γ. Hence, the extremal function for the absolute value of the coefficient γ1 is f=fν,γ,1, and those for the absolute value of the coefficients γ2 and γ3 in the first relations are given as f=fν,γ,2 and f=fν,γ,3, respectively, whereas in the second relations, it is given as f=fν,γ,1.By takingγ=1 and ν=c−1/2 in Theorem 14, we have the following result which is the estimates obtained by Ponnusamy et al. for c∈2,3 in Theorem 2.7 of [27].Corollary 15.
Letf∈FOν. Then,
(78)γ1≤c4,γ2≤4c+c248,γ3≤2c+c248.All the inequalities are sharp.Theorem 16.
Letf∈FOν,γ, then
(79)an≤∏k=2nk−2+γ2ν+1n!n=2,3,⋯.Proof.
Since(80)ϕz=2ν+121+z1−zγ−2ν−12with B1=γ2ν+1 is convex, the result is obtained by utilization of Lemma 13.Remark 17.
Takingγ=1 in Theorem16, the result presented in [3], Theorem 18 is obtained.Theorem 18.
Letf∈FOν,γ. Then, we have sharp bounds for complex μ:
(81)a3−μa22≤γ22v+162v+1−32v+12,forμ−4v+132v+1≥23γ2v+1,γ2v+16,forμ−4v+132v+1<23γ2v+1.Proof.
Letf∈FOν,γ; then, from (11), by the concept of the subordination, there is ω∈Ω with ωz=∑n=1∞wnzn such that
(82)1+zf″zf′z=ϕωz=1+B1w1z+B1w2+B2w12z2+⋯.
So, we obtain(83)2a2=B1w1,6a3−4a22=B1w2+B2w12.
Therefore,(84)a3−μa22=16B1w2+γw12.
The outcomes are given by applying Lemma12 with γ=B2/B1+B11−3μ/2 and (65). Equality occurs in the first inequality by f=fν,γ,1 and in the second inequality for f=fν,γ,2.


## 4. Conclusion

In the current study, we investigate some geometric features such as strongly starlikeness and close-to-convexity for the classFOν,γ. Further, sharp bounds are given on Fekete-Szegö functionals and logarithmic coefficients for functions belonging to this class.



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