Zalcman Functional and Majorization Results for Certain Subfamilies of Holomorphic Functions

Abstract

In this paper, we investigate sharp coefficient functionals, like initial four sharp coefficient bounds, sharp Fekete–Szegö functionals, and, for $n=1$ and 2, sharp Zalcman functionals are evaluated for class of functions associated with tangent functions. Furthermore, we provide some majorization results for some non-vanishing holomorphic functions, whose ratios are related to various domains in the open unit disk.

1. Introduction

Here, in this section, we provide some fundamental and significant concepts for a better comprehension of the primary results. Starting with the most fundamental definition, for which we use the symbol $\mathcal{A}$. Let $\mathcal{A}$ stand for the family of analytic functions with the following series representation:

$$\omega \left(\kappa \right)=\kappa +a_2{\kappa }^2+a_3{\kappa }^3+a_4{\kappa }^4+\cdots =\kappa +\sum _{\ell =2}^{\infty }{d}_n{\kappa }^n,$$

(1)

where

$$\omega \left(0\right)={\omega }^{\prime }\left(0\right)-1=0\mathrm{and}z\in \mathsf{\Omega }:=\{\kappa :\kappa \in \mathbb{C}\mathrm{and}|\kappa |<1\},$$

$\mathbb{C}$ being the set of complex numbers. Additionally, all univalent functions of family $\mathcal{A}$ are included in the separate family $\mathcal{S}$. In 1916, Biberbach [1] made the coefficient conjecture, which helped the topic gain popularity as a potential area for further study, although the concept of function theory was first developed in 1851. This conjuncture was proven in 1985 by De-Branges [2]. Many of the top academics in the world tried to support or refute the Bieberbach conjucture between 1916 and 1985. As a result, they discovered several subfamilies of $\mathcal{S}$ family of normalised univalent functions that are connected to various image domains. The most fundamental and important subclasses of the functions class $\mathcal{S}$ are the ${\mathcal{S}}^{*}$ and $\mathcal{K}$, known as families of starlike and convex functions, respectively. Analytically,

$${\mathcal{S}}^{*}=\left\{\omega \in \mathcal{S}:\mathrm{Re}\left(\frac{\kappa {\omega }^{\prime }\left(\kappa \right)}{\omega \left(\kappa \right)}\right)>0\right\}$$

and

$$\mathcal{K}=\left\{\omega \in \mathcal{S}:\mathrm{Re}\left(\frac{{\left(\kappa {\omega }^{\prime }\left(\kappa \right)\right)}^{\prime }}{{\omega }^{\prime }\left(\kappa \right)}\right)>0\right\}.$$

The concept of quasi-subordination between holomorphic functions was first introduced by Robertson [3] in 1970. Let $g_1$ and $g_2$ are two members of the family $\mathcal{A}$, are connected to quasi-subordination relationship, mathematically demonstrated by $g_1{\prec }_q{g}_2,$ if there occurs functions $\mathsf{\Phi },u\in \mathcal{A}$, so that $\frac{\kappa {\omega }^{\prime }\left(\kappa \right)}{\mathsf{\Phi }\left(\kappa \right)}$ is holomorphic in $\mathsf{\Omega }$ with properties

$$\left|\mathsf{\Phi }\left(\kappa \right)\right|\le 1,\left|u\left(\kappa \right)\right|\le \left|\kappa \right|\mathrm{and}u\left(0\right)=0,$$

satisfying the relationship

$$g_1\left(\kappa \right)=\mathsf{\Phi }\left(\kappa \right)g_2\left(u\left(\kappa \right)\right),\kappa \in \mathsf{\Omega }.$$

Moreover, by choosing

$$u\left(\kappa \right)=\kappa \mathrm{and}\mathsf{\Phi }\left(\kappa \right)=1,$$

we get, at the most, helpful ideas in GFT, known as subordination between holomorphic functions. Actually, if $g_2\in \mathcal{S}$, then for $g_1,g_2\in \mathcal{A}/$, the relationship of subordination has

$$g_1\left(\kappa \right)\prec g_2\left(\kappa \right)\iff \left[g_1\left(\mathsf{\Omega }\right)\subset g_2\left(\mathsf{\Omega }\right)\mathrm{with}{g}_1\left(0\right)=g_2\left(0\right)\right].$$

By assuming that $u\left(\kappa \right)=\kappa$, the above definition reduced to the majorization between holomorphic functions and is given by

$$g_1\left(\kappa \right)\ll g_2\left(\kappa \right),g_1\left(\kappa \right),g_2\left(\kappa \right)\in \mathcal{A}/.$$

So, $g_1\left(\kappa \right)\ll g_2\left(\kappa \right),$ if the function $\mathsf{\Phi }\left(\kappa \right)\in \mathcal{A}$ meets the requirement $\left|\mathsf{\Phi }\left(\kappa \right)\right|\le 1,$ such that

$$g_1\left(\kappa \right)=\mathsf{\Phi }\left(\kappa \right)g_2\left(\kappa \right),\kappa \in \mathsf{\Omega }.$$

(2)

In 1967, MacGregor [4] first proposed this concept. This concept has appeared in numerous articles. For some recent research on this subject, we refer the readers to see [5,6,7,8,9].

In 1992, Ma and Minda defined [10]

$${\mathcal{S}}^{*}\left(\varphi \right)=\left\{\omega \in \mathcal{A}:\frac{\kappa {\omega }^{{}^{\prime }}\left(\kappa \right)}{\omega \left(\kappa \right)}\prec \varphi \left(\kappa \right)\right\}.$$

(3)

with $\Re \left(\varphi \right)>0$ in $\mathsf{\Omega }$, additionally, the function $\varphi$ maps $\mathsf{\Omega }$ onto a star-shaped region, and the image domain is symmetric about the real axis and starlike, with respect to $\varphi \left(0\right)=1$ with $\varphi {}^{\prime }\left(0\right)>0$. The set ${\mathcal{S}}^{*}\left(\varphi \right)$ generalizes a number of subfamilies of the function class $\mathcal{A}$, including, for instance:

• If

  $$\varphi \left(\kappa \right)=\frac{1+L\kappa }{1+M\kappa },$$

  with $-1\le M<L\le 1$, then

  $${\mathcal{S}}^{*}[L,M]\equiv {\mathcal{S}}^{*}\left(\frac{1+L\kappa }{1+M\kappa }\right),$$

  where ${\mathcal{S}}^{*}[L,M]$ is the class of Janowski starlike functions, see [11].
• Choose $\varphi \left(\kappa \right)=\sqrt{1+\kappa },$ we receive the family ${\mathcal{S}}_{\mathcal{L}}^{*},$ defined and investigated by Sokol et al. [12].
• For the function

  $$\varphi \left(\kappa \right)=1+{sinh}^{-1}\kappa ,$$

  we receive the class ${\mathcal{S}}_{\rho }^{*},$ introduced by kumar and Arora [13].
• If $\varphi \left(\kappa \right)=e^{\kappa },$ then the class ${\mathcal{S}}^{*}\left(\varphi \right)$ becomes ${\mathcal{S}}_e^{*}$, which is defined and studied by Mendiratta [14].
• For $\varphi \left(\kappa \right)=1+sin\left(\kappa \right),$ the class ${\mathcal{S}}^{*}\left(\varphi \right)$ reduces to the class ${\mathcal{S}}_{sin}^{*}$. The family ${\mathcal{S}}_s^{*}$ was introduced by Cho et al. [15] as:

  $${\mathcal{S}}_{sin}^{*}=\left\{\omega \in \mathcal{A}:\frac{\kappa {\omega }^{\prime }\left(\kappa \right)}{\omega \left(\kappa \right)}\prec 1+sin\left(\kappa \right),(\kappa \in \mathsf{\Omega })\right\},$$

  (4)

  which means that $\frac{\kappa {\omega }^{\prime }\left(\kappa \right)}{\omega \left(\kappa \right)}$ lies in an eight shaped region.
• If we pick $\varphi \left(\kappa \right)=cos\left(\kappa \right),$ we receive the family ${\mathcal{S}}_{cos}^{*},$ initiated by Bano and Raza [16].
• If we select $\varphi \left(\kappa \right)=sech\left(\kappa \right),$ we obtain a family ${\mathcal{S}}_{sec h}^{*}$ introduced by Al-Shbeil et al. [17].

In this paper, we define the following subfamily of holomorphic functions

$${\mathcal{S}}_{tan}^{*}=\left\{\omega \in \mathcal{A}:\frac{\kappa {\omega }^{\prime }\left(\kappa \right)}{\omega \left(\kappa \right)}\prec \frac{2+tan\left(\kappa \right)}{2}\right\}(\kappa \in \mathsf{\Omega }).$$

(5)

2. Set of Lemma’s

The following are some useful lemma, which we use in our main finding.

Let $\mathcal{P}$ denote the family of all holomorphic functions p with a positive real part, having the following series representation:

$$p\left(\kappa \right)=1+\sum _{n=1}^{\infty }{c}_n{\kappa }^n,\kappa \in \mathsf{\Omega }.$$

(6)

Lemma 1.

If $p\in \mathcal{P}$, then the following estimates are valid:

$$\begin{array}{ccc}\hfill \left|c_k\right|& \le & 2,k\ge 1,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \left|c_{k+n}-\mu c_k{c}_n\right|& <& 2,0<\mu \le 1,\hfill \end{array}$$

(8)

and, for $\eta \in \mathbb{C}$, we have

$$\left|c_2-\eta c_1^2\right|<2max\left\{1,\left|2\eta -1\right|\right\}.$$

(9)

For the inequalities (7) and (8), see [18], and (9) is given in [19].

Lemma 2

([20]). If $p\in \mathcal{P}$ and has the form (6), then

$$|{\alpha }_1{c}_1^3-{\alpha }_2{c}_1{c}_2+{\alpha }_3{c}_3|\le 2|{\alpha }_1|+2|{\alpha }_2-2{\alpha }_1|+2|{\alpha }_1-{\alpha }_2+{\alpha }_3|,$$

(10)

where ${\alpha }_1,{\alpha }_2$, and ${\alpha }_3$ are real numbers.

Lemma 3

([21]). Let $m_1,n_1,l_1$, and $r_1$ satisfy the inequalities $m_1,r_1\in \left(0,1\right)$ and

$$\begin{array}{cc}\hfill & 8r_1\left(1-r_1\right)\left[{\left(m_1{n}_1-2l_1\right)}^2+{\left(m_1\left(r_1+m_1\right)-n_1\right)}^2\right]\hfill \\ \hfill & +m_1\left(1-m_1\right){\left(n_1-2r_1{m}_1\right)}^2\hfill \\ \hfill & \le 4m_1^2{\left(1-m_1\right)}^2{r}_1\left(1-r_1\right).\hfill \end{array}$$

If $h\in \mathcal{P}$ and is of the form (6), then

$$\left|l_1{c}_1^4+r_1{c}_2^2+2m_1{c}_1{c}_3-\frac{3}{2}{n}_1{c}_1^2{c}_2-c_4\right|\le 2.$$

3. Coefficient Bounds and Zalcman Functional for the Family ${\mathcal{S}}_{tan}^{*}$

One of the established conjectures in the study of Geometric function theory, posited by Lawrence Zalcman in 1960, states that the coefficients of class $\mathcal{S}$ satisfy the inequality

$$\left|d_n^2-d_{2n-1}\right|\le {\left(n-1\right)}^2.$$

The equality in the above inequality holds only for the renowned Koebe function $k\left(\kappa \right)=\frac{\kappa }{{\left(1-\kappa \right)}^2}$ and its rotations. The famous Fekete–Szegö inequality holds for $n=2.$ Zalcman’s conjecture in its original setting, i.e., for the entire class $\mathcal{S}$, has been proved by Krushkal for $n=3,4,5,$ and 6 in [22]. This important result is now presented also in the books [23,24]. For detailed studies on Zalcman functionals, see the articles [25,26,27,28,29,30].

Theorem 1.

Let $\omega \in {\mathcal{S}}_{tan}^{*}.$ Then, the following estimates hold:

$$\begin{array}{ccc}\hfill \left|d_2\right|& \le & \frac{1}{2},\hfill \\ \hfill \left|d_3\right|& \le & \frac{1}{4},\hfill \\ \hfill \left|d_4\right|& \le & \frac{1}{6},\hfill \\ \hfill \left|d_5\right|& \le & \frac{1}{8}.\hfill \end{array}$$

These estimates are sharp for the function

$${\omega }_n\left(\kappa \right)=\kappa +\frac{1}{2n}{\kappa }^{n+1}+\cdots ,n=1,2,3,\dots .$$

(11)

Proof. 

Let $\omega \in {\mathcal{S}}_{tan}^{*}.$ Then, by property of the Schwarz function $u\left(\kappa \right),$ so that $u\left(0\right)=0$ and $\left|u\left(\kappa \right)\right|\le \left|\kappa \right|,$ we have

$$\frac{\kappa {\omega }^{\prime }\left(\kappa \right)}{\omega \left(\kappa \right)}=\frac{2+tan\left(u\left(\kappa \right)\right)}{2}.$$

As there is one–one correspondence between $u\left(\kappa \right)$ and the function with the positive real part $p\left(\kappa \right),$ we can write

$$u\left(\kappa \right)=\frac{p\left(\kappa \right)-1}{p\left(\kappa \right)+1}.$$

Now,

$$\begin{array}{ccc}\hfill \frac{2+tan\left(u\left(\kappa \right)\right)}{2}& =& 1+\frac{1}{4}{c}_1\kappa +\left(\frac{1}{4}{c}_2-\frac{1}{8}{c}_1^2\right){\kappa }^2+\left(\frac{1}{12}{c}_1^3-\frac{1}{4}{c}_2{c}_1+\frac{1}{4}{c}_3\right){\kappa }^3\hfill \\ & & \left(-\frac{1}{16}{c}_1^4+\frac{1}{4}{c}_1^2{c}_2-\frac{1}{4}{c}_3{c}_1-\frac{1}{8}{c}_2^2+\frac{1}{4}{c}_4\right){\kappa }^4+\cdots .\hfill \end{array}$$

(12)

And,

$$\begin{array}{ccc}\hfill \frac{\kappa {\omega }^{\prime }\left(\kappa \right)}{\omega \left(\kappa \right)}& =& 1+d_2\kappa +\left(2d_3-d^2\right){\kappa }^2+\left(3d_4-3d_2{d}_3-d_2^3\right){\kappa }^3+\hfill \\ & & \left(4d_5-d_2^4+4d_2^2{d}_3-4d_2{d}_4-2d_3^2\right)+\cdots .\hfill \end{array}$$

(13)

On comparing (12) and (13), we have

$$\begin{array}{ccc}\hfill d_2& =& \frac{1}{4}{c}_1,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill d_3& =& \frac{1}{8}{c}_2-\frac{1}{16}{c}_1^2,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill d_4& =& \frac{1}{144}{c}_1^3-\frac{5}{96}{c}_2{c}_1+\frac{1}{12}{c}_3,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill d_5& =& -\frac{65}{9216}{c}_1^4+\frac{13}{384}{c}_1^2{c}_2-\frac{1}{24}{c}_3{c}_1-\frac{3}{128}{c}_2^2+\frac{1}{16}{c}_4.\hfill \end{array}$$

(17)

Applying (7) to (14), we have

$$\left|d_2\right|\le \frac{1}{2}.$$

To find the bound of $d_3$, apply (8) to (15), and then we have

$$\left|d_3\right|\le \frac{1}{4}.$$

Applying (10) to (16), we get

$$\left|d_4\right|\le \frac{1}{6}.$$

And,

$$\left|d_5\right|=\frac{1}{16}\left|\frac{65}{576}{c}_1^4-\frac{13}{24}{c}_1^2{c}_2+\frac{2}{3}{c}_3{c}_1+\frac{3}{8}{c}_2^2-c_4\right|.$$

Now, using Lemma 3 with $l_1=\frac{65}{576},$ $r_1=\frac{3}{8},$ $m_1=\frac{1}{3}$, and $n_1=\frac{13}{36},$ we have

$$\left|d_5\right|\le \frac{1}{8}.$$

□

Theorem 2.

Let $\omega \in {\mathcal{S}}_{tan}^{*}.$ Then, for a complex number $\lambda ,$ we have

$$\left|d_3-\lambda d_2^2\right|\le \frac{1}{4}max\left\{1,\left|\lambda \right|\right\}.$$

The estimate is sharp for function ${\omega }_1$, defined by (11).

Proof. 

From (14) and (15), we have

$$\left|d_3-\lambda d_2^2\right|=\frac{1}{8}\left|c_2-\frac{1+\lambda }{2}{c}_1^2\right|.$$

The desired result is achieved by applying (9) to the above equation. □

Set $\lambda =1$ in the above Theorem 2, and then we receive the following corollary, which is a special case of Zalcman functional when $n=2.$

Corollary 1.

Let $\omega \in {\mathcal{S}}_{tan}^{*}.$ Then,

$$\left|d_3-d_2^2\right|\le \frac{1}{4}.$$

A sharp result is achieved for the function ${\omega }_1$, defined by (11).

Proof. 

From (15) and (17), we have

$$\begin{array}{ccc}\hfill \left|d_3^2-d_5\right|& =& \left|\frac{101}{9216}{c}_1^4-\frac{19}{384}{c}_1^2{c}_2+\frac{1}{24}{c}_3{c}_1+\frac{5}{128}{c}_2^2-\frac{1}{16}{c}_4\right|\hfill \\ & \le & \frac{1}{8}(\mathrm{using}\mathrm{Lemma}3).\hfill \end{array}$$

Which is our desired proof. For sharpness, set $n=4$ in Equation (11), that is

$${\omega }_4\left(\kappa \right)=\kappa +\frac{1}{8}{\kappa }^4+\cdots .$$

And,

$$\left|d_3^2-d_5\right|=\left|d_5\right|=\frac{1}{8}.$$

□

4. Majorization Results

First, we select the holomorphic nonvanishing functions ${\mathsf{\Lambda }}_1$ and ${\mathsf{\Lambda }}_2$ in open unit disc $\mathsf{\Omega }$ with conditions

$${\mathsf{\Lambda }}_1\left(0\right)=1\mathrm{and}{\mathsf{\Lambda }}_2\left(0\right)=1.$$

Following that, the families established in this article consist of functions $\omega \in \mathcal{A}$, whose ratios $\frac{\omega \left(\kappa \right)}{\kappa q\left(\kappa \right)}$ and $q\left(\kappa \right)$ are, respectively, subordinated to ${\mathsf{\Lambda }}_1$ and ${\mathsf{\Lambda }}_2,$ for the certain holomorphic function q, with $q\left(0\right)=1$ as

$$\frac{\omega \left(\kappa \right)}{\kappa q\left(\kappa \right)}\prec {\mathsf{\Lambda }}_1\left(\kappa \right)\mathrm{and}q\left(\kappa \right)\prec {\mathsf{\Lambda }}_2\left(\kappa \right).$$

We will now choose a certain particular functions rather than ${\mathsf{\Lambda }}_1$ and ${\mathsf{\Lambda }}_2$. These options include

$${\mathsf{\Lambda }}_1\left(\kappa \right)=cos\left(\kappa \right),$$

or

$${\mathsf{\Lambda }}_1\left(\kappa \right)=\sqrt{1+\kappa },$$

or

$${\mathsf{\Lambda }}_1\left(\kappa \right)=sech\left(\kappa \right),$$

and

$${\mathsf{\Lambda }}_2\left(\kappa \right)=\frac{2+tan\left(\kappa \right)}{2}.$$

Using the previously discussed concepts, we now study the following new subfamilies:

$$\begin{array}{ccc}\hfill {\mathsf{{\rm Y}}}_{cos}& =& \left\{\omega \in \mathcal{A}:\frac{\omega \left(\kappa \right)}{\kappa q\left(\kappa \right)}\prec cos\left(\kappa \right)\mathrm{and}q\left(\kappa \right)\prec \frac{2+tan\left(\kappa \right)}{2}\right\},\hfill \\ \hfill {\mathsf{{\rm Y}}}_{\mathcal{L}}& =& \left\{\omega \in \mathcal{A}:\frac{\omega \left(\kappa \right)}{\kappa q\left(\kappa \right)}\prec \sqrt{1+\kappa }\mathrm{and}q\left(\kappa \right)\prec \frac{2+tan\left(\kappa \right)}{2}\right\},\hfill \\ \hfill {\mathsf{{\rm Y}}}_{sech}& =& \left\{\omega \in \mathcal{A}:\frac{\omega \left(\kappa \right)}{\kappa q\left(\kappa \right)}\prec sech\left(\kappa \right)\mathrm{and}q\left(\kappa \right)\prec \frac{2+tan\left(\kappa \right)}{2}\right\}.\hfill \end{array}$$

Majorization results for each of the aforementioned families are covered in the current section.

Lemma 4.

Let $q\left(\kappa \right)\prec 1+\frac{tan\left(\kappa \right)}{2}.$ Then, for $\left|\kappa \right|\le s,$ we have

$$\left|\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}\right|\le \frac{s{sec}^2\left(s\right)}{\left(1-s^2\right)\left(2-tan\left(s\right)\right)}.$$

Proof. 

As $q\left(\kappa \right)\prec 1+\frac{tan\left(\kappa \right)}{2},$ by the properties of the Schwarz function $u\left(\kappa \right),$ we have

$$q\left(\kappa \right)=\frac{2+tan\left(u\left(\kappa \right)\right)}{2}.$$

Taking log differentiation on both sides, we have

$$\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}=\frac{u^{\prime }\left(\kappa \right){sec}^2\left(u\left(\kappa \right)\right)}{2+tan\left(u\left(\kappa \right)\right)}$$

(18)

Let us assume that $u\left(\kappa \right)={\mathrm{Re}}^{i\theta },$ $\theta \in \left[0,2\pi \right]$, and $\left|\kappa \right|=R\le s.$ Consider

$$\begin{array}{ccc}\hfill {\left|tan\left({\mathrm{Re}}^{i\theta }\right)\right|}^2& =& {\left(\frac{sin\left(Rcos\theta \right)cos\left(Rcos\theta \right)}{{cos}^2\left(Rcos\theta \right)+{sinh}^2\left(Rcos\theta \right)}\right)}^2+\left(\frac{sinh\left(Rsin\theta \right)cosh\left(Rsin\theta \right)}{{cos}^2\left(Rcos\theta \right)+{sinh}^2\left(Rcos\theta \right)}\right)\hfill \\ & =& Q\left(\theta \right)\left(\mathrm{say}\right).\hfill \end{array}$$

Clearly, $Q\left(\theta \right)$ is satisfied with the condition of the even function, which is $Q\left(-\theta \right)=Q\left(\theta \right),$ so we could consider interval $\left[0,\pi \right]$ instead of $\left[0,2\pi \right].$ Moreover, $Q^{\prime }\left(\theta \right)=0$ has three roots in interval $0\le \theta \le \pi$, namely, $\theta =0,\frac{\pi }{2}$ and $\pi .$ We notice that

$$Q\left(\frac{\pi }{2}\right)={tanh}^{2}R\mathrm{and}Q\left(0\right)={tan}^{2}R=Q\left(\pi \right).$$

Moreover, we see that

$$\begin{array}{ccc}\hfill max\left\{Q\left(0\right),Q\left(\frac{\pi }{2}\right),Q\left(\pi \right)\right\}& =& {tan}^{2}R\le {tan}^{2}s,\hfill \\ \hfill min\left\{Q\left(0\right),Q\left(\frac{\pi }{2}\right),Q\left(\pi \right)\right\}& =& {tanh}^{2}R\le {tanh}^{2}s.\hfill \end{array}$$

Hence,

$$tanhs\le \left|tan\left({\mathrm{Re}}^{i\theta }\right)\right|\le tans.$$

(19)

Also,

$$\begin{array}{ccc}\hfill {\left|sec\left({\mathrm{Re}}^{i\theta }\right)\right|}^2& =& {\left(\frac{cos\left(Rcos\theta \right)cosh\left(Rsin\theta \right)}{{cos}^2\left(Rcos\theta \right)+{sinh}^2\left(Rsin\theta \right)}\right)}^2+{\left(\frac{sin\left(Rcos\theta \right)sinh\left(Rsin\theta \right)}{{cos}^2\left(Rcos\theta \right)+{sinh}^2\left(Rsin\theta \right)}\right)}^2\hfill \\ & =& E\left(\theta \right)\left(\mathrm{say}\right).\hfill \end{array}$$

As $E\left(-\theta \right)=E\left(\theta \right),$ it is an even function; therefore, we have to consider interval $\left[0,\pi \right].$ Furthermore, $E^{\prime }\left(\theta \right)=0$ also has three roots: $\theta =0,\frac{\pi }{2}$, and $\pi ,$

$$E\left(\frac{\pi }{2}\right)=sec{h}^{2}R\mathrm{and}Q\left(0\right)={sec}^{2}R=E\left(\pi \right).$$

Moreover, we see that

$$\begin{array}{ccc}\hfill max\left\{E\left(0\right),E\left(\frac{\pi }{2}\right),E\left(\pi \right)\right\}& =& {sec}^{2}R\le {sec}^{2}s,\hfill \\ \hfill min\left\{E\left(0\right),E\left(\frac{\pi }{2}\right),E\left(\pi \right)\right\}& =& sec{h}^{2}R\le sec{h}^{2}s.\hfill \end{array}$$

Hence,

$$sec{h}^{2}s\le {\left|sec\left({\mathrm{Re}}^{i\theta }\right)\right|}^2\le {sec}^{2}s.$$

(20)

Also, the Schwarz function $u\left(\kappa \right)$ satisfies the following condition

$$\left|u^{\prime }\left(\kappa \right)\right|\le \frac{1-{\left|u\left(\kappa \right)\right|}^2}{1-{\left|\kappa \right|}^2}=\frac{1-R^2}{1-s^2}\le \frac{1}{1-s^2}.$$

(21)

Now, from (18), we have

$$\left|\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}\right|\le \frac{\left|\kappa \right|\left|u^{\prime }\left(\kappa \right)\right|\left|{sec}^2\left(u\left(\kappa \right)\right)\right|}{2-\left|tan\left(u\left(\kappa \right)\right)\right|}.$$

(22)

Now, using (19)–(21) to above inequality (22), we receive the desired achievement. □

Theorem 3.

Let $\omega \in \mathcal{A}$, $g\in {\mathsf{{\rm Y}}}_{cos}$ and also assume that $\omega \left(\kappa \right)\ll g\left(\kappa \right)$ in Ω. Then, for $\left|\kappa \right|\le s_1$

$$\left|{\omega }^{\prime }\left(\kappa \right)\right|\le \left|g^{\prime }\left(\kappa \right)\right|,$$

where $s_1$ is the smallest positive root of the equation

$$\left(\left(1-2s-s^2\right)coss-ssinhs\right)\left(2-tans\right)-scoss{sec}^{2}s=0.$$

Proof. 

Let $g\in {\mathsf{{\rm Y}}}_{cos}.$ Then, by the property of the Schwarz function $u\left(\kappa \right)$, we receive

$$\frac{g\left(\kappa \right)}{\kappa q\left(\kappa \right)}=cos\left(u\left(\kappa \right)\right).$$

After some straightforward calculations, we receive

$$\frac{\kappa g^{\prime }\left(\kappa \right)}{g\left(\kappa \right)}=1+\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}-\frac{\kappa u^{\prime }\left(\kappa \right)sin\left(u\left(\kappa \right)\right)}{cos\left(u\left(\kappa \right)\right)}.$$

Now, by using (19)–(21) along with Lemma 4, we have

$$\begin{array}{ccc}\hfill \left|\frac{g\left(\kappa \right)}{g^{\prime }\left(\kappa \right)}\right|& =& \frac{\left|\kappa \right|}{\left|1+\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}-\frac{\kappa u^{\prime }\left(\kappa \right)sin\left(u\left(\kappa \right)\right)}{cos\left(u\left(\kappa \right)\right)}\right|}\hfill \\ & \le & \frac{s\left(1-s^2\right)\left(2-tans\right)coss}{\left(1-s^2\right)\left(2-tans\right)coss-s{sec}^{2}scoss-ssinhs\left(2-tans\right)}.\hfill \end{array}$$

(23)

From (2), we can write

$$\omega \left(\kappa \right)=\mathsf{\Phi }\left(\kappa \right)g\left(\kappa \right),$$

upon differentiating, we get

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\kappa \right)& =& {\mathsf{\Phi }}^{\prime }\left(\kappa \right)g\left(\kappa \right)+\mathsf{\Phi }\left(\kappa \right)g^{\prime }\left(\kappa \right)\hfill \\ & =& \left({\mathsf{\Phi }}^{\prime }\left(\kappa \right)\frac{g\left(\kappa \right)}{g^{\prime }\left(\kappa \right)}+\mathsf{\Phi }\left(\kappa \right)\right)g^{\prime }\left(\kappa \right).\hfill \end{array}$$

(24)

Also, the Schwarz function satisfies the following condition

$$\left|{\mathsf{\Phi }}^{\prime }\left(\kappa \right)\right|\le \frac{1-{\left|\mathsf{\Phi }\left(\kappa \right)\right|}^2}{1-{\left|\kappa \right|}^2}=\frac{1-{\left|\mathsf{\Phi }\left(\kappa \right)\right|}^2}{1-s^2}.$$

(25)

Now, using (23) and (24) in (25), we obtain

$$\left|{\omega }^{\prime }\left(\kappa \right)\right|\le \left(\frac{s\left(1-{\left|\mathsf{\Phi }\left(\kappa \right)\right|}^2\right)\left(2-tans\right)coss}{\left(1-s^2\right)\left(2-tans\right)coss-s{sec}^{2}scoss-ssinhs\left(2-tans\right)}+\left|\mathsf{\Phi }\left(\kappa \right)\right|\right)\left|g^{\prime }\left(\kappa \right)\right|,$$

let us assume that $\left|\mathsf{\Phi }\left(\kappa \right)\right|=\delta ,$ where $0\le \delta \le 1.$ Then, the above inequality becomes

$$\left|{\omega }^{\prime }\left(\kappa \right)\right|\le \mathsf{\Psi }\left(\delta ,s\right)\left|g^{\prime }\left(\kappa \right)\right|,$$

where

$$\mathsf{\Psi }\left(\delta ,s\right)=\frac{s\left(1-{\delta }^2\right)\left(2-tans\right)coss}{\left(1-s^2\right)\left(2-tans\right)coss-s{sec}^{2}scoss-ssinhs\left(2-tans\right)}+\delta .$$

To determine $s_1,$ it is sufficient to choose

$$s_1=max\left(s\in \left[0,1\right]:\mathsf{\Psi }\left(\delta ,s\right)\le 1,\mathrm{for}\mathrm{all}\delta \in \left[0,1\right]\right),$$

or

$$s_1=max\left(s\in \left[0,1\right]:G\left(\delta ,s\right)\ge 0,\mathrm{for}\mathrm{all}\delta \in \left[0,1\right]\right),$$

where

$$G\left(\delta ,s\right)=\left(\left(1-s^2-s\left(1+\delta \right)\right)coss-ssinhs\right)\left(2-tans\right)-s{sec}^{2}scoss.$$

Clearly, if we select $\delta =1,$ then we can see that the function $G\left(\delta ,s\right)$ gets its minimum value, which is

$$min\left(G\left(\delta ,s\right),s\in \left[0,1\right]\right)=G\left(\delta ,1\right)=H\left(s\right),$$

where

$$H\left(s\right)=\left(\left(1-s^2-2s\right)coss-ssinhs\right)\left(2-tans\right)-s{sec}^{2}scoss.$$

Next, we have the following inequalities:

$$H\left(0\right)=2>0\mathrm{and}H\left(1\right)=-2.8492<0.$$

There is indeed a $s_1,$ so that $H\left(s\right)\ge 0,$ for every s in $\left[0,s_1\right],$ $s_1$ is the smallest positive root of the equation

$$\left(\left(1-s^2-2s\right)coss-ssinhs\right)\left(2-tans\right)-s{sec}^{2}scoss.$$

The proof is completed. □

Theorem 4.

Let $\omega \in \mathcal{A}$, $g\in {\mathsf{{\rm Y}}}_{\mathcal{L}}$ and also suppose that $\omega \left(\kappa \right)\ll g\left(\kappa \right)$ in Ω. Then, for $\left|\kappa \right|\le s_1$,

$$\left|{\omega }^{\prime }\left(\kappa \right)\right|\le \left|g^{\prime }\left(\kappa \right)\right|,$$

where $s_1$ is the smallest positive root of the equation

$$\left(2-5s-3s^2\right)\left(2-tans\right)-2s{sec}^{2}s=0.$$

Proof. 

If $g\in {\mathsf{{\rm Y}}}_{\mathcal{L}}.$ Then, a holomorphic function $u\left(\kappa \right)$ such that $u\left(0\right)=0$ and $\left|u\left(\kappa \right)\right|\le \left|\kappa \right|,$ so that

$$\frac{g\left(\kappa \right)}{\kappa q\left(\kappa \right)}=\sqrt{1+u\left(\kappa \right)}.$$

After, simple calculations, we attain

$$\frac{\kappa g^{\prime }\left(\kappa \right)}{g\left(\kappa \right)}=1+\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}-\frac{\kappa u^{\prime }\left(\kappa \right)}{2\left(1+u\left(\kappa \right)\right)}.$$

Now, by using (21), we have

$$\begin{array}{ccc}\hfill \left|\frac{\kappa u^{\prime }\left(\kappa \right)}{2\left(1+u\left(\kappa \right)\right)}\right|& \le & \frac{\left|\kappa \right|\left(1+\left|u\left(\kappa \right)\right|\right)}{2\left(1-\left|u\left(\kappa \right)\right|\right)}\le \frac{\left|\kappa \right|\left(1+\left|\kappa \right|\right)}{2\left(1-{\left|\kappa \right|}^2\right)}=\frac{\left|\kappa \right|}{2\left(1-\left|\kappa \right|\right)}\hfill \\ & \le & \frac{s}{2\left(1-s\right)}.\hfill \end{array}$$

(26)

In the light of (26) and Lemma 4, we have

$$\begin{array}{ccc}\hfill \left|\frac{g\left(\kappa \right)}{g^{\prime }\left(\kappa \right)}\right|& =& \frac{\left|\kappa \right|}{\left|1+\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}-\frac{\kappa u^{\prime }\left(\kappa \right)}{2\left(1+u\left(\kappa \right)\right)}\right|}\hfill \\ & \le & \frac{\left|\kappa \right|}{1-\left|\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}\right|-\left|\frac{\kappa u^{\prime }\left(\kappa \right)}{2\left(1+u\left(\kappa \right)\right)}\right|}\hfill \\ & \le & \frac{2s\left(1-s^2\right)\left(2-tans\right)}{2\left(1-s^2\right)\left(2-tans\right)-2s{sec}^{2}s-s\left(1+s\right)\left(2-tans\right)}.\hfill \end{array}$$

(27)

From (2), we can write

$$\omega \left(\kappa \right)=\mathsf{\Phi }\left(\kappa \right)g\left(\kappa \right),$$

upon differentiating, we get

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\kappa \right)& =& {\mathsf{\Phi }}^{\prime }\left(\kappa \right)g\left(\kappa \right)+\mathsf{\Phi }\left(\kappa \right)g^{\prime }\left(\kappa \right)\hfill \\ & =& \left({\mathsf{\Phi }}^{\prime }\left(\kappa \right)\frac{g\left(\kappa \right)}{g^{\prime }\left(\kappa \right)}+\mathsf{\Phi }\left(\kappa \right)\right)g^{\prime }\left(\kappa \right).\hfill \end{array}$$

(28)

Also, Schwarz function satisfies the following condition

$$\left|{\mathsf{\Phi }}^{\prime }\left(\kappa \right)\right|\le \frac{1-{\left|\mathsf{\Phi }\left(\kappa \right)\right|}^2}{1-{\left|\kappa \right|}^2}=\frac{1-{\left|\mathsf{\Phi }\left(\kappa \right)\right|}^2}{1-s^2}.$$

(29)

Now, using (27) and (29) in (28), we obtain

$$\left|{\omega }^{\prime }\left(\kappa \right)\right|\le \left(\frac{2s\left(1-{\left|\mathsf{\Phi }\left(\kappa \right)\right|}^2\right)\left(2-tans\right)}{2\left(1-s^2\right)\left(2-tans\right)-2s{sec}^{2}s-s\left(1+s\right)\left(2-tans\right)}+\left|\mathsf{\Phi }\left(\kappa \right)\right|\right)\left|g^{\prime }\left(\kappa \right)\right|,$$

By using the same calculations as in Theorem 4, the desired outcomes are attained. □

Theorem 5.

Let $\omega \in \mathcal{A}$, $g\in$Υ${}_{sec h}$ and also presume that $\omega \left(\kappa \right)\ll g\left(\kappa \right)$ in Ω. Then, for $\left|\kappa \right|\le s_1$,

$$\left|{\omega }^{\prime }\left(\kappa \right)\right|\le \left|g^{\prime }\left(\kappa \right)\right|,$$

where $s_1$ is the smallest positive root of the equation

$$\left(1-s^2-s\left(2+tans\right)\right)\left(2-tans\right)-s{sec}^{2}s=0.$$

Proof. 

If $g\in {\mathsf{{\rm Y}}}_{sec h}$, then a holomorphic function $u\left(\kappa \right)$ is achieved, such that $u\left(0\right)=0$, $\left|u\left(\kappa \right)\right|\le \left|\kappa \right|,$ and

$$\frac{g\left(\kappa \right)}{\kappa q\left(\kappa \right)}=sech\left(u\left(\kappa \right)\right).$$

After simple calculations, we attain

$$\frac{\kappa g^{\prime }\left(\kappa \right)}{g\left(\kappa \right)}=1+\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}-\kappa u^{\prime }\left(\kappa \right)tanh\left(u\left(\kappa \right)\right).$$

Let $u\left(\kappa \right)={\mathrm{Re}}^{i\theta },$ $\left|\kappa \right|=s\le s_1$. Then,

$$\begin{array}{ccc}\hfill {\left|tanh\left({\mathrm{Re}}^{i\theta }\right)\right|}^2& =& {\left(\frac{sinh\left(Rcos\theta \right)cosh\left(Rcos\theta \right)}{{sinh}^2\left(Rcos\theta \right)+{cos}^2\left(Rsin\theta \right)}\right)}^2+{\left(\frac{sin\left(Rsin\theta \right)cos\left(Rsin\theta \right)}{{sinh}^2\left(Rcos\theta \right)+{cos}^2\left(Rsin\theta \right)}\right)}^2\hfill \\ & =& \mathcal{ϝ}\left(\theta \right)\left(\mathrm{say}\right).\hfill \end{array}$$

Clearly, $\mathcal{ϝ}\left(\theta \right)$ is even function, so it is enough to consider interval $\left[0,\pi \right]$ instead of $\left[0,\pi \right]$ for $\theta .$ Also, after simple calculations, ${\mathcal{ϝ}}^{\prime }\left(\theta \right)=0$ has three roots, namely, $\theta =0,\pi$ and $\frac{\pi }{2}.$ We notice that

$$\mathcal{ϝ}\left(\frac{\pi }{2}\right)={tan}^{2}R\mathrm{and}\mathcal{ϝ}\left(0\right)={tanh}^{2}R=\mathcal{ϝ}\left(\pi \right).$$

Moreover, we see that

$$max\left\{\mathcal{ϝ}\left(0\right),\mathcal{ϝ}\left(\frac{\pi }{2}\right),\mathcal{ϝ}\left(\pi \right)\right\}={tan}^{2}R.$$

Hence,

$$\left|tanh\left({\mathrm{Re}}^{i\theta }\right)\right|\le tanR\le tans.$$

(30)

In the light of (21), (30), and Lemma 4, we have

$$\begin{array}{ccc}\hfill \left|\frac{g\left(\kappa \right)}{g^{\prime }\left(\kappa \right)}\right|& =& \frac{\left|\kappa \right|}{\left|1+\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}-\frac{\kappa u^{\prime }\left(\kappa \right)}{2\left(1+u\left(\kappa \right)\right)}\right|}\hfill \\ & \le & \frac{\left|\kappa \right|}{1-\left|\frac{\kappa q^{\prime }\left(\kappa \right)}{q\left(\kappa \right)}\right|-\left|\kappa u^{\prime }\left(\kappa \right)tanh\left(u\left(\kappa \right)\right)\right|}\hfill \\ & \le & \frac{s\left(1-s^2\right)\left(2-tans\right)}{\left(1-s^2\right)\left(2-tans\right)-s{sec}^{2}s-stans\left(2-tans\right)}.\hfill \end{array}$$

(31)

From (2), we can write

$$\omega \left(\kappa \right)=\mathsf{\Phi }\left(\kappa \right)g\left(\kappa \right),$$

and upon differentiating, we receive

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(\kappa \right)& =& {\mathsf{\Phi }}^{\prime }\left(\kappa \right)g\left(\kappa \right)+\mathsf{\Phi }\left(\kappa \right)g^{\prime }\left(\kappa \right)\hfill \\ & =& \left({\mathsf{\Phi }}^{\prime }\left(\kappa \right)\frac{g\left(\kappa \right)}{g^{\prime }\left(\kappa \right)}+\mathsf{\Phi }\left(\kappa \right)\right)g^{\prime }\left(\kappa \right).\hfill \end{array}$$

(32)

Now using (31) and (25) in (32), we obtain

$$\left|{\omega }^{\prime }\left(\kappa \right)\right|\le \left(\frac{s\left(1-{\left|\mathsf{\Phi }\left(\kappa \right)\right|}^2\right)\left(2-tans\right)}{\left(1-s^2\right)\left(2-tans\right)-s{sec}^{2}s-stans\left(2-tans\right)}+\left|\mathsf{\Phi }\left(\kappa \right)\right|\right)\left|g^{\prime }\left(\kappa \right)\right|.$$

The same calculations as in Theorem 4 are used to produce the required outcomes. □

5. Conclusions

In this article, we, first, derived sharp coefficient estimates, sharp Fekete–Szegö, and Zalcman functionals for the subfamily of holomorphic functions associated with tangent functions. Furthermore, we looked into the majorization results for a some families of holomorphic functions that are associated with various shapes domains. It is possible to extend these results into more subfamiles, including, for example, for the class of meromorphic functions and the families of harmonic functions. Furthermore, one may attempt to produce results for certain subclasses of q-starlike functions: for details, see [31,32,33,34,35,36,37,38,39,40,41].