Majorization results for non vanishing analytic functions in different domains

1.   Introduction

For better understanding of the main results, here in this part of the paper we give some basic and important concepts. We start from the very basic definition, which we denote by symbol $\mathfrak{A}$. The class $\mathfrak{A}$ consists of all analytic functions in the open disk

and if $f\left(z\right)$ is contained in $\mathfrak{A}$, the relations

are satisfied. In addition, the family $\mathfrak{S\subset A}$ includes all univalent functions. The coefficient conjecture stated by Biberbach ^[1] in 1916 contributed to the field's emergence as a viable area of future research, despite the fact that function theory was formed in 1851. De-Branges ^[2] proved this conjecture in 1985. Between 1916 and 1985, many of the world's leading scholars attempted to confirm or reject the Bieberbach conjecture. As a result, they found a number of subclasses of the $\mathfrak{S}$ family of normalised univalent functions that are linked to different image domains. The $\mathfrak{S}^{\ast }$ and $\mathcal{K}$, classes of starlike and convex functions, respectively, are the most fundamental and important subclasses of the functions class $\mathfrak{S}$, which are described as

and

Robertson ^[3] established the concept of quasi-subordination between holomorphic functions in $1970$. Two functions $\mathcal{F} _{1}\left(z\right), \mathcal{F}_{1}\left(z\right) \in \mathfrak{A}$ are related to the relationship of quasi-subordination, indicated mathematically by $\mathcal{F}_{1}\left(z\right) \prec _{q}\mathcal{F}_{2}\left(z\right),$ if there exist functions $\phi \left(z\right), u\left(z\right) \in \mathfrak{A}$ so that $\frac{zf^{\prime }\left(z\right) }{\phi \left(z\right) }$ is regular in $\mathfrak{D}$ with

obeying the relationship

Furthermore, by selecting

we gain one of the most useful geometric function theory ideas known as subordination between regular functions. In fact, if $\mathcal{F}_{2}\left(z\right) \in \mathfrak{S, }$ then, for $\mathcal{F}_{1}\left(z\right), \mathcal{F}_{2}\left(z\right) \in \mathfrak{A}$, the subordination relationship has

By taking $u\left(z\right) = z,$ the above definition of quasi-subordination becomes the majorization between holomorphic functions and is written mathematically by

That is

if the function $\phi \left(z\right) \in \mathfrak{A}$ having the condition $\left \vert \phi \left(z\right) \right \vert \leqq 1$ such that

MacGregor ^[4] developed this concept in 1967. Several papers have been written in which this concept has been utilized. The work of Srivastava and Altintas ^[5], Goyal and Goswami ^[6,7], Cho et al. ^[8], Li et al. ^[9], Aouf and Prajapat ^[10], Goswami and Aouf ^[11], El-Ashwah and Panigraht ^[12] and Tang et al. ^[13,14] are worth noting on this subject. For some recent study on this topic, we refer the readers to see ^{[15,16,17,18,19]}.

Ma and Minda ^[20] examined the general form of the family $\mathfrak{S}^{\ast }$ in 1992, and it was given by

where $\Lambda \left(z\right)$ is analytic function having the condition $\Lambda ^{\prime }\left(0\right) > 0$ and in $\mathfrak{D}$ its real part is greater than 0. In addition, with regard to $\Lambda \left(0\right) = 1$, the function $\Lambda \left(z\right)$ maps $\mathfrak{D}$ onto a star-like shaped area. Ma and Minda ^[20] investigated on growth, distortion, and covering theorems, along with other aspects. Various sub-families of the normalized holomorphic class $\mathfrak{A}$ have been explored as a particular example of class $\mathfrak{S}^{\ast }\left(\Lambda \right)$ in recent years. Some of them are listed below:

$({\rm{i}})$ Choosing $\Lambda (z)$ as

then we obtain the class given by

where $\mathfrak{S}^{\ast }[M, N]$ is the functions class define in ^[21], see also ^[22,23]. Furthermore, the class $\mathfrak{S}^{\ast }\left(\zeta \right)$ given by

where $\mathfrak{S}^{\ast }\left(\zeta \right)$ is the class of starlike function of order $\zeta$.

$({\rm{ii}})$ The following class:

was studied in ^[24] by Stankiewicz and Sokól.

$({\rm{iii}})$ By taking $\Lambda (z) = 1+\sin z,$ the family $\mathfrak{S} ^{\ast }(\Lambda (z))$ leads to the class $\mathfrak{S}_{\sin }^{\ast },$ which was investigated by Cho et al. ^[25]. On the other hand, the function class given by

was studied in ^[26] (see also ^[27]).

$({\rm{iv}})$ The class $\mathfrak{S}_{R}^{\ast }\equiv \mathfrak{S} ^{\ast }\left(\Lambda (z)\right)$ with $\Lambda (z) = 1+\frac{z}{J}\frac{J+z }{J-z},$ $J = 1+\sqrt{2}$ is studied in ^[28]. While the following families:

and

were considered, respectively, by Abdullah et.al ^[29] and Bano and Raza ^[30].

For some more recent and interesting investigations on some subclasses of analytic and bi-univalent functions, we may refer the readers to see ^{[31,32,33,34]}.

Now, we choose the nonvanishing holomorphic functions $h_{1}\left(z\right)$ and $h_{2}\left(z\right)$ in $\mathfrak{D}$ with

Then, for the classes which we described in this article, contain such function $f\left(z\right) \in \mathfrak{A}$ whose ratios $\frac{f\left(z\right) }{ zq\left(z\right) }$ and $q\left(z\right)$ are subordinated to $h_{1}\left(z\right)$ and $h_{2}\left(z\right),$ respectively, for certain holomorphic function $q\left(z\right)$ with $q\left(0\right) = 1$ as

Instead of $h_{1}\left(z\right)$ and $h_{2}\left(z\right),$ we will now select certain specific functions. These choices are

or

or

or

and

We now investigate the following new subfamilies by using the above-mentioned concepts:

We will examine majorization problems for each of the above-mentined families in this article: $\mathcal{F}_{\cos }, \mathcal{F}_{\mathfrak{S} \mathcal{L}}, \mathcal{F}_{\exp }$ and $\mathcal{F}_{car}$.

2.   Main results

To proved majorization results for the families $\mathcal{F}_{\cos }, \mathcal{F}_{\mathfrak{S}\mathcal{L}}, \mathcal{F}_{\exp }$ and $\mathcal{F} _{car},$ we need the following lemmas.

Lemma 2.1. Let $q(z)\prec 1+\tanh z$ and $\left \vert z\right \vert \leq r$. Then, $q\left(z\right)$ satisfies the following conditions:

and

Proof. If $q(z)\prec 1+\tanh z,$ then

for certain Schwartz function $u(z)$. After a very simple computations, we now get

Let $u(z) = R(e^{it}),$ with $\left \vert z\right \vert = r\leq R,$ $-\pi \leq \theta \leq \pi.$ Upon certain simple computation, we get

where

Now, we can write

So, we have

Let us suppose

A simple computation give us that $0$, $\pm \frac{\pi }{2}$ and $\pm \pi$ are the zeros of $\Psi^{\prime} (\theta)$ in $[-\pi, \pi]$. We observe that

Furthermore, we see that

Hence,

Similarly, we demonstrate that

Now, from well-known inequality for schwarz function $u(z),$ we attain

Now using $(2.7)$–$(2.9)$ in $(2.3),$ we get $(2.2).$

Lemma 2.2. Suppose that $q(z)\prec 1+\sin z$ ($\left \vert z\right \vert \leq r$). Then $q$ satisfies the following conditions:

and

Proof. For proof see ^[35].

Theorem 2.1. Let $f\left(z\right) \in \mathfrak{A},$ $g\in \mathcal{F}_{\cos }$ and also assume that $f\left(z\right) \ll g\left(z\right)$ in $\mathfrak{D }$. Then, for $\left \vert z\right \vert \leq r_{1},$

where $r_{1}$ is the smallest positive root of the equation

Proof. If $g\in \mathcal{F}_{\cos },$ then by the subordination relationship, we get

Now, by some simple computations, we obtain

Now by using $\left(2.7\right)$–$\left(2.9\right)$ and $\left(2.11\right)$ along with Lemma 2.1, we have

From $(1.2),$ we can write

Differentiating $\left(2.15\right)$ on both sides, we obtain

Also, the Schwarz function $\phi \left(z\right)$ satisfies the following inequality:

Now applying $(2.14)$ and $\left(2.17\right)$ in $(2.16),$ we attain

which by putting

becomes the inequality

where

To determine $r_{1},$ it is sufficient to choose

or, equivalently,

where

Obviously, if we choose $\eta = 1,$ then we can see that the function $\Phi _{1}\left(r, \eta \right)$ gets its minimum value, namely,

where

Next, we have the following inequalities:

There is indeed a $r_{1}$ so that $\Phi _{1}\left(r\right) \geq 0$ for every $r\in \left[0, r_{1}\right],$ where $r_{1}$ is the smallest positive root of Eq $\left(2.12\right)$. Thus, we have completed the proof of our result.

Theorem 2.2. Let $f\left(z\right) \in \mathfrak{A},$ $g\in \mathcal{F}_{ \mathfrak{S}\mathcal{L}}$ and also assume that $f\left(z\right)$ is majorized by $g\left(z\right)$ in $\mathfrak{D}.$ Then, for $\left \vert z\right \vert \leq r_{2}$,

where $r_{2}$ is the smallest positive root of the equation

Proof. If $g\in \mathcal{F}_{\mathfrak{S}\mathcal{L}}.$ Then, a holomorphic function $u\left(z\right)$ in $\mathfrak{D}$ occurs with $u\left(0\right) = 0$ and $\left \vert u\left(z\right) \right \vert \leq \left \vert z\right \vert$ so that

After some simple computations, we now have

Utilizing $\left(2.9\right),$ we get

By virtue of $\left(2.9\right)$ and Lemma 2.1, we get

Now, by using $\left(2.21\right)$ and $\left(2.17\right)$ in $\left(2.16\right),$ we get

The required results are obtained by the same computations as in Theorem 2.1, along with the use of $\left(2.18\right).$

Theorem 2.3. Let $f\left(z\right) \in \mathfrak{A},$ $g\left(z\right) \in \mathcal{F}_{\exp }$ and also assume that $f\left(z\right) \ll g\left(z\right)$ in $\mathfrak{D}$. Then, for $\left \vert z\right \vert \leq r_{3}$,

where $r_{3}$ is the smallest positive root of the equation

Proof. If $g\in \mathcal{F}_{\exp }$. Then, by subordination relationship, we have

Now, after some easy computations, we obtain

Now by using $\left(2.7\right)$–$\left(2.9\right)$ in conjunction with Lemma 2.1, we have

Now, by using $\left(2.24\right)$ and $\left(2.17\right)$ in $\left(2.16\right),$ we get

The required results is obtained by the same computations as in Theorem 2.1, along with the use of $\left(2.18\right).$

Theorem 2.4. Let $f\left(z\right) \in \mathfrak{A},$ $g\left(z\right) \in \mathcal{F}_{car}$ and also assume that $f\left(z\right) \ll g\left(z\right)$ in $\mathfrak{D}$. Then, for $\left \vert z\right \vert \leq r_{4}$,

where $r_{4}$ is the smallest positive root of the equation

with

Proof. If $g\in \mathcal{F}_{car}.$ Then by subordination relationship, we have

After some simple computations, we now have

Now, by using $\left(2.7\right)$–$\left(2.9\right),$ we have

where

Now, by using $\left(2.27\right)$ and $\left(2.17\right)$ in $\left(2.16\right),$ we get

We obtain the results directly utilizing the same calculation which are presented in the proof of our Theorem 2.1 in conjunctions with $\left(2.18\right).$

3.   Conclusions

We investigated on majorization problems for a certain subfamilies of regular functions that are connected to distinct shapes domains. Other subfamilies of these problems can be investigated, such as the families of meromorphic functions and the families of harmonic functions. One may attempt the suggested results for different subclasses of $q$-starlike functions.

Acknowledgement

The first author (Huo Tang) was partly supported by the Natural Science Foundation of China under Grant 11561001, the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant NJYT18-A14, the Natural Science Foundation of Inner Mongolia of China under Grants 2022MS01004 and 2020MS01011, and the Higher School Foundation of Inner Mongolia of China under Grant NJZY20200, the Program for Key Laboratory Construction of Chifeng University (No. CFXYZD202004), the Research and Innovation Team of Complex Analysis and Nonlinear Dynamic Systems of Chifeng University (No. cfxykycxtd202005) and the Youth Science Foundation of Chifeng University (No. cfxyqn202133).

Conflict of interest

The authors declare that they have no conflicts of interest.