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

The purpose of the current paper is to investigate some geometric properties of the class F O ð ν , γ Þ , called strongly Ozaki close-to-convex functions, such as strongly starlikeness and close-to-convexity. Further, we ﬁ nd sharp bounds on Fekete-Szegö functionals and logarithmic coe ﬃ cients for functions belonging to the class F O ð ν , γ Þ , which incorporates some known outcomes as the speci ﬁ c cases.


Introduction and Preliminaries
Let U denote the open unit dick in the complex plane ℂ. Let A be the class of functions f of the following normalized form: 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 ωð0Þ = 0 and jωðzÞj < 1 for z ∈ U, i.e., Ω, is considered as the family of Schwarz functions.
For two analytic functions f and F in the open unit dick U, it is said that the function f is subordinate to the function F in U, written f ðzÞ ≺ FðzÞ, if there exists a Schwarz function ω such that f ðzÞ = FðωðzÞÞ for all z ∈ U. In particular, if the function F is univalent in U, the following equivalence holds: We denote by S * ðαÞ the subclass of A consisting of all f ∈ A for which f is a starlike of order α, with and denote by KðαÞ the subclass of A consisting of all f ∈ A for which f is a convex of order α, with Also, the subclassS * ðαÞ of a strongly starlike function of order in U is defined as Note thatS * ð1Þ = S * ð0Þ = S * and Kð0Þ = K are the class of starlike functions in U and the class of convex functions in U, respectively. Furthermore, CðαÞ 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 and we denote byCðαÞ the subclass of A consisting of all of f ∈ A for which Individually, Cð0Þ = C is the class of close-to-convex functions in U andCð1Þ =C is the subclass of close-toconvex functions in U (see [1]). Here, we understand that Argw is a number in ð−π, π: Recently, many authors have studied the families of analytic functions of the class A and also investigated bound estimation problems, geometric property issues, and related topics for functions belonging to these families in [2][3][4][5][6][7][8][9][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 * ðφÞ of starlike functions, which was defined by Ma and Minda [11]. Also, they investigated the majorization issue for the various subclasses S * ðφÞ for different special functions φ. Moreover, estimates for the coefficients of majorized functions regarding the class S * ðφÞ were given. Further, Alimohammadi et al. [2] introduced a subclass of A and extended the class GðαÞðα ∈ ð0, 1Þ, defined by Nunokawa et al. in [12], consisting of all f ∈ A satisfying and studied some geometric properties like close-toconvexity and strongly starlikeness. They determined sharp bounds of Fekete-Szegö functionals and logarithmic coefficients for this class. Kargar and Ebadian [9] considered the subclass FðνÞ of locally univalent functions f ∈ A in U satisfying the inequality: for some −1/2 < ν ≤ 1.
Recently, Allu et al. [3], motivated essentially by the subclass FðνÞ, introduced the class F O ðν, γÞ and obtained sharp bounds for three first coefficients and the corresponding inverse coefficients for the functions of this class. Definition 1 [3]. Let γ ∈ ð0, 1 and ν ∈ ½1/2, 1. Then, f is called strongly Ozaki close-to-convex if and only if Note that the class F O ðν, 1Þ = F O ðνÞ was introduced in [9] and members of this class were called Ozaki close-to-convex functions. Also, F O ð1, 1Þ = Fð1Þ was studied by Ponnusamy et al. [13]. Furthermore, F O ð1/2, 1Þ = K.
It is remarkable that by means of the principle of subordination between analytic functions, the definition of the class F O ðν, γÞ can be rewritten as follows: The present paper was undertaken to investigate some geometric features of the class F O ðν, γÞ such as close-toconvexity and strongly starlikeness. In addition, we found estimates for the coefficients a n and give sharp bounds on Fekete-Szegö functionals and logarithmic coefficients for functions belonging to the class F O ðν, γÞ, which incorporates some known outcomes as the specific cases.

Some Geometric Properties of the Class F O ðν, γÞ
In this section, we investigate some geometric properties like strongly starlikeness and close-to-convexity for the class F O ðν, γÞ to present the relation of this class with the wellknown 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 by Q the class of all complex-valued functions q for which q is univalent at each U \ EðqÞ and q′ðξÞ ≠ 0 for all ξ ∈ ∂U \ EðqÞ where Lemma 2 ([16], Lemma 2.2d (i)). Let q ∈ Q with qð0Þ = a and let pðzÞ = a + p n z n + ⋯ be analytic in U with pðzÞ ≡ 1 and n ≥ 1. If p is not subordinate to q in U, then there exist z 0 ∈ U and ξ 0 ∈ ∂U \ EðqÞ such that fpðzÞ: z ∈ U, jzj < jz 0 jg ⊂ qðUÞ: Lemma 3 (see [14,15]). Let the function p given by be analytic in U with pð0Þ = 1 and pðzÞ ≠ 0 for all z ∈ U: If there exists a point z 0 ∈ U with Journal of Function Spaces for some β > 0, then where then Proof. The result is proven by contradiction. Let f ∈ A and define the function M : Then, it is concluded that M is analytic in U, Mð0Þ = 1, and where M 1 is analytic in U with M 1 ðz 0 Þ ≠ 0: Then, Hence, with z ⟶ z 0 , in the right hand of the above equality, the argument can properly take any value between −π and π, which contradicts to (20).

Journal of Function Spaces
Then, h is a differentiable function on ð0, aÞ, and Now, we define gðβÞ as Then, gð0Þ So, the function g has a negative value on ð0, β 0 Þ and a positive value on ðβ 0 , 1Þ.
Next, let ArgðMðz 0 ÞÞ = −ðβπ/2Þ. Then, we write Mðz 0 Þ = a β ðcos ðβπ/2Þ − i sin ðβπ/2ÞÞ. Thus, for k ≤ −1 and utilizing (37), we get which contradicts to (20). From the above contradictions, it results in Hence, the proof is completed. ∈ UÞ and so it is well known that f is univalent. then Proof. The result is proven by contradiction. To prove our result, we set the function M : U ⟶ ℂ by

Journal of Function Spaces
Then, M is analytic in U, pð0Þ = 1, MðzÞ ≠ 0 for all z ∈ U, and If there is a point z 0 ∈ U, then with jzj < jz 0 j, and Then, from Lemma 3, we have where ½Mðz 0 Þ 1/β = ±ia ða > 0Þ and k is stated by (17) or (18). For the case ArgðMðz 0 ÞÞ = απ/2when and k ≥ 1, we have which contradicts to (41). Next, for the case ArgðMðz 0 ÞÞ = −ðαπ/2Þ when and k ≤ −1, by applying the same method mentioned above, it can be concluded that which contradicts to (41).
As a result, from the above contradictions, we obtain and therefore, the proof is completed.

Coefficient Bounds
In this section, we find sharp bounds on Fekete-Szegö functionals and logarithmic coefficients (see [17][18][19][20][21][22]) for functions belonging to the class F O ðν, γÞ. 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 γ n of f ∈ S are defined by the next form: These coefficients are of great significance for different estimates in the theory of univalent functions (see [19][20][21]).

Journal of Function Spaces
Lemma 11 [23,24]. If ω ∈ Ω with ωðzÞ = ∑ ∞ n=1 w n z n ðz ∈ UÞ, then, the following sharp estimate is given: for any real numbers q 1 and q 2 where the sets A, B, C, D are stated as follows: Lemma 12 [23,25]. Let ω ∈ Ω with ωðzÞ = ∑ ∞ n=1 w n z n for all z ∈ U. Then, The result is sharp for the function ωðzÞ = z 2 or ωðzÞ = z.
Lemma 13 [26]. Let φ be a convex function in U with the form φðzÞ = 1 + ∑ ∞ n=1 B n z n : If function f ∈ KðφÞ, then a n j j ≤ Theorem 14. Let f ∈ F O ðν, γÞ. Then, ! , All the inequalities are sharp.

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

Data Availability
No data were used to support this study.