Coefficient Estimates in a Class of Close-to-Convex Functions

Abstract

In this paper, we consider analytic functions with the formalization $f\left(0\right)=f^{\prime }\left(0\right)-1=0$, which satisfy $\mathrm{Re}\left\{(1-\alpha z^2)f^{\prime }\left(z\right)\right\}>0,z\in \Delta =\{z\in \mathbb{C}:|z|<1\},$ where $\alpha \in [-1,1]$. The set of such functions is a subclass of the class of close-to-convex functions. In this paper, we present sharp bounds of $|a_n|$, where $a_n$ is the $n-$th Taylor’s coefficients for functions in this class. In addition, we consider several symmetric coefficient problems when the second coefficient $a_2$ is established. In particular, we provide bounds of $|a_{n+1}-a_n|$ and of $|2n{a}_{2n}-(2n-1)a_{2n-1}|$ for the considered class under this additional assumption.

1. Introduction and Definitions

The class of the functions f analytic in $\Delta =\{z\in \mathbb{C}:|z|<1\}$ with the series expansion

$$f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n,z\in \Delta$$

(1)

will be denoted by $\mathcal{A}$. Note that, for $f\in \mathcal{A}$, we have $f\left(0\right)=0=f^{\prime }\left(0\right)-1$. Next, let $\mathcal{S}$ be the class of functions $f\in \mathcal{A}$ that are univalent in $\Delta$.

Let $\mathcal{P}$ be the Carathéodory class, that is, the class of analytic functions of the form

$$q\left(z\right)=1+\sum _{n=1}^{\infty }{p}_n{z}^n,z\in \Delta$$

(2)

with $\mathrm{Re}\left\{q\left(z\right)\right\}>0,z\in \Delta$.

Recall that $f\in \mathcal{A}$ is said to be a starlike function if $\mathrm{Re}\left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right)>0,z\in \Delta$, while a function $f\in \mathcal{A}$ is called convex when $z{f}^{\prime }$ is starlike. We will use the standard notations $S^{\ast }$ and $\mathcal{K}$ for the classes of starlike and convex functions, respectively. A function $f\in \mathcal{A}$ is said to be a close-to-convex function if there exists a convex function g and $\beta \in \mathbb{R}$ such that

$$\mathrm{Re}\left(e^{i\beta }{\displaystyle \frac{f^{\prime }\left(z\right)}{g^{\prime }\left(z\right)}}\right)>0,z\in \Delta .$$

The class of all the close-to-convex functions will be denoted by $\mathcal{C}$.

In 1985, de Branges [1] published the proof of the famous conjecture, known since 1916 as the Bieberbach conjecture, which states that the coefficients of a function $f\in \mathcal{S}$ with the Taylor series expansion (1) satisfy the inequality

$$|a_n| \le n$$

(3)

for every positive integer n. The equality in (3) holds only for the Koebe function

$$k\left(z\right)={\displaystyle \frac{z}{{(1-z)}^2}},$$

or for its rotations. The Bieberbach conjecture, extraordinary in its simplicity, turned out to be a difficult problem. For decades, it posed a formidable challenge to mathematicians, and work on this problem resulted in the discovery of many new, valuable methods and techniques in geometric function theory. Progressive published articles provided proofs of the Bieberbach conjecture for a fixed n or for important subclasses of the class $\mathcal{S}$ of univalent functions. More details on this topic can be found, for example, in [2].

One of the topics that arose from the attempts to prove the Bieberbach conjecture was the problem of estimating the functional related to successive coefficients, that is, $|a_{n+1}| - |a_n|$, for the class $\mathcal{S}$ of univalent functions. The interest in this topic stemmed from the fact that the estimate $\left|\right|a_{n+1}| - |a_n\left|\right| \le 1$ for $f\in \mathcal{S}$ implies $|a_n| \le n$. Let us recall several results in this direction. In 1963, Hayman [3] proved that

$$\left|\right|a_{n+1}| - |a_n\left|\right| \le A,n=1,2,\dots$$

holds in the class $\mathcal{S}$, where A is an absolute constant. Next, in 1978, Leung [4] demonstrated the proof of the estimation

$$\left|\right|a_{n+1}| - |a_n\left|\right| \le 1,n=1,2,3,\dots$$

for $f\in {\mathcal{S}}^{\ast }$, which was conjectured by Pommerenke [5]. The equality in this condition only holds for the function $f\left(z\right)={\displaystyle \frac{z}{(1-\gamma z)(1-\zeta z)}}$ for some $\gamma$ and $\zeta$ that satisfy $\left|\gamma \right|=\left|\zeta \right|=1.$ In 2017, Li and Sugawa [6] obtained the following sharp upper estimate

$$|a_{n+1}| - |a_n| \le 1/(n+1),n=2,3,\dots$$

for the class $\mathcal{K}$ of convex functions. Moreover, for $n=2,3$, they provided the sharp lower bounds $1/2$ and $1/3$, respectively.

Another direction of research, resulting from the dependence

$$\left|\right|a_{n+1}| - |a_n\left|\right| \le |a_{n+1}-a_n|,$$

is related to the problem of estimating the modulus of the difference of successive coefficients in various subclasses of the class $\mathcal{S}$. Let us quote here some results. In 1981, Robertson [7] proved that if $f\in \mathcal{K}$, then the following inequality holds for each integer $n\ge 2$:

$$|a_{n+1}-a_n| \le {\displaystyle \frac{2n+1}{3}}|a_2-1|.$$

(4)

In particular, for $n=2$ or $n=3$, we obtain from (4) the following:

$$|a_3-a_2| \le {\displaystyle \frac{5(2-\frac{a_2}{2})}{6}}\mathrm{and}|a_4-a_3| \le {\displaystyle \frac{7(2-\frac{a_2}{2})}{6}}.$$

(5)

In the abovementioned article [6], Li and Sugawa considered the class $\mathcal{K}\left(p\right),p=a_2/2$ of convex functions with the second coefficient $a_2$ fixed, and they proved that, for $f\in \mathcal{K}\left(p\right)$ with $p\in [0,2]$, the following estimates hold:

$$|a_3-a_2| \le {\displaystyle \frac{5(2p+1)(2-p)}{6}}$$

and

$$|a_4-a_3| \le \left\{\begin{array}{cc}{\displaystyle \frac{p^3+50p^2-64p+64}{192}}\hfill & \mathrm{when}0 \le p<{\displaystyle \frac{8}{7}},\hfill \\ {\displaystyle \frac{-3p^3+4p^2+6p-4}{12}}\hfill & \mathrm{when}{\displaystyle \frac{8}{7}} \le p \le 2.\hfill \end{array}\right.$$

The above results improve those given in (5). In 2010, Brown [8] proved the following estimation for the Carathéodory class $\mathcal{P}$:

$$|p_{n+1}-p_n| \le 2\sqrt{2-\mathrm{Re}\left\{p_1\right\}}$$

with equality for $p\left(z\right)={\displaystyle \frac{1+e^{i\alpha }z}{1-e^{i\alpha }z}}$, where $\alpha ={cos}^{-1}\left({\displaystyle \frac{b}{2}}\right)$ and $\mathrm{Re}\left\{p_1\right\}=2b.$ Quite recently, Zaprawa and Trąbka-Więcław [9] researched the class of analytic functions defined by

$$\mathrm{Re}\left\{(1-z^2)f^{\prime }\left(z\right)\right\}>0,z\in \Delta$$

with an additional condition $f^{″}\left(0\right)=p,p\in [-2,2]$. Among other things they gave the following sharp estimates for this class:

$$|a_3-a_2| \le \left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}(2-p)\hfill & \mathrm{when}p\in [-2,1],\hfill \\ {\displaystyle \frac{1}{6}}(2-p)(2p+1)\hfill & \mathrm{when}p\in [1,2],\hfill \end{array}\right.$$

and the estimates

$$|a_4-a_3| \le \left\{\begin{array}{cc}{\displaystyle \frac{1}{36}}(38-17p\hfill & \mathrm{when}p\in [-2,5/3],\hfill \\ {\displaystyle \frac{1}{12}}(2-p)(3p^2+2p-2)\hfill & \mathrm{when}p\in [5/3,2],\hfill \end{array}\right.$$

which are sharp for $p\in [5/3,2]$. More results on successive coefficients can be found in [2] (pp. 113–115) (see also [10,11,12,13,14]).

Motivated by these ideas and results, we will focus on the problem of maximizing selected coefficient functionals for certain classes of close-to-convex functions. Let us start with the definition.

For a given $-1 \le \alpha \le 1$, consider the subclass of $\mathcal{A}$ consisting of functions that satisfy

$$\mathrm{Re}\left\{(1-\alpha z^2)f^{\prime }\left(z\right)\right\}>0,z\in \Delta .$$

(6)

Condition (6) can be rewritten in the following equivalent form:

$$\mathrm{Re}\left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{F_{\alpha }\left(z\right)}}\right)>0,z\in \Delta ,$$

(7)

where the function

$$F_{\alpha }\left(z\right)={\displaystyle \frac{z}{1-\alpha z^2}}=z+\sum _{n=2}^{\infty }{\alpha }^n{z}^{2n+1},z\in \Delta$$

is starlike for $\alpha \in [-1,1]$. This family was introduced in [15] and is denoted by ${\mathcal{C}}_0\left(F_{\alpha }\right)$. Taking into account (7), we can observe that ${\mathcal{C}}_0\left(F_{\alpha }\right)\subset \mathcal{C}$.

If $\alpha \in [-1,1]$ is fixed, then we obtain the known classes of analytic functions. For example, we can obtain ${\mathcal{C}}_0\left(F_0\right)=\mathcal{R}$, where the symbol $\mathcal{R}$ stands for the class of bounded turning (see [16] (p. 101), [17]). For $\alpha =-1$, we obtain the class ${\mathcal{C}}_0(z/(1+z^2))$ defined by the condition

$$\mathrm{Re}\left\{(1+z^2)f^{\prime }\left(z\right)\right\}\ge 0,z\in \Delta .$$

We found, in [15], a sharp estimate of $|a_3-a_2|$ for the functions of this class with the second coefficient fixed. If we put $\alpha =1$ in (6), then

$$\mathrm{Re}\left\{(1-z^2)f^{\prime }\left(z\right)\right\}>0,z\in \Delta$$

defines the class that was introduced and investigated by Zaprawa and Trąbka-Więcław in [9]. In this paper, the authors found that this family of functions has an interesting symmetry property, that is, if $f\in {\mathcal{C}}_0(z/(1-z^2))$, then $\overline{f\left(\overline{z}\right)}\in {\mathcal{C}}_0(z/(1-z^2))$ also, and the images of $\Delta$ under these functions are symmetric with respect to the real axis.

Let us also mention the related family of functions $f\in \mathcal{A}$, for which, for some $|{\zeta }_1| = |{\zeta }_2| = 1$, the following condition holds:

$$\mathrm{Re}\left\{e^{i\beta }(1-{\zeta }_{1}z)(1-{\zeta }_{2}z)f^{\prime }\left(z\right)\right\}\ge 0,z\in \Delta ,$$

which was introduced by Lecko [18] (see also [19,20,21]).

Condition (6), which defines the family ${\mathcal{C}}_0\left(F_{\alpha }\right)$, gives a very useful correspondence between this class and the Carathéodory class, namely

$$f\in {\mathcal{C}}_0\left(F_{\alpha }\right)\iff (1-\alpha z^2)f^{\prime }\left(z\right)\in \mathcal{P},$$

(8)

or, equivalently, for each $q\in \mathcal{P}$, there exists a function f in the class ${\mathcal{C}}_0\left(F_{\alpha }\right)$ such that

$$(1-\alpha z^2)f^{\prime }\left(z\right)=q\left(z\right).$$

(9)

Then, for f with Expansion (1) and the q of the form (2), the following relationships hold among the coefficients of these functions:

$$2a_2=p_1,n{a}_n-\alpha (n-2)a_{n-2}=p_{n-1},n=3,4,5,\dots .$$

It is well known that if $q\in \mathcal{P}$, then $|p_n| \le 2,n=1,2,3,\dots$. From this, using the equality $2a_2=p_1$, it follows that $|2a_2| = |p_1| \le 2$ and, consequently, $|f^{″}\left(0\right)| \le 2$ or $|a_2| \le 1$.

In this paper, we were looking for the bounds of some coefficient functionals in ${\mathcal{C}}_0\left(F_{\alpha }\right)$, partially with the additional assumption that $a_2=p/2$ is fixed. In [15], we introduced the subclass of ${\mathcal{C}}_0\left(F_{\alpha }\right)$, denoted by ${\mathcal{C}}_0(F_{\alpha },p)$, which consists of functions whose second coefficient in series expansion is fixed as $a_2=p/2$ and $p\in [-2,2]$. It should be noted that this family can be defined as follows:

$${\mathcal{C}}_0(F_{\alpha },p)=\{f\in {\mathcal{C}}_0\left(F_{\alpha }\right):f^{″}\left(0\right)=p\}.$$

Observe that if $f\in {\mathcal{C}}_0(F_{\alpha },p)$, then $a_2=p/2$ and $p_1=p$ for the function q connected to f by Equality (9). The second coefficient was fixed to obtain additional normalization for the function f at the point $z=0$, that is, we fixed $f^{″}\left(0\right)$. In this way, we reduced our considerations to a class of functions with fixed $f\left(0\right)$, $f^{\prime }\left(0\right)$, and $f^{″}\left(0\right)$ together.

2. Preliminaries

We begin with the Carathéodory-Toeplitz theorem (see [22]).

Lemma 1

(Carathéodory-Toeplitz theorem). Let q be of the form $q\left(z\right)=1+\sum _{n=1}^{\infty }{p}_n{z}^n,$ with $p_n\in \mathbb{C}$. Then, $q\in \mathcal{P}$, if and only if the so-called Toeplitz determinants

$$D_n=\left|\begin{array}{ccccc}2& p_1& p_2& \dots & p_n\\ p_{-1}& 2& p_1& \dots & p_{n-1}\\ ⋮& ⋮& ⋮& & ⋮\\ p_{-n}& p_{-n+1}& p_{-n+2}& \dots & 2\end{array}\right|,$$

with $p_{-j}=\overline{p_j}$ ($j\ge 1$), are non-negative for all $n\ge 1$. Moreover, if $D_j>0$, for each $j=1,2,\dots ,k-1$, and if $D_k=0$, then the function q has the following form:

$$q\left(z\right)=\sum _{j=1}^k{\gamma }_j{\displaystyle \frac{1+{ϵ}_{j}z}{1-{ϵ}_{j}z}},{\gamma }_j>0,\left|{ϵ}_j\right|=1,{ϵ}_m\ne {ϵ}_n\mathrm{for}m\ne n.$$

(10)

It should be noted that $q\left(0\right)=1$ results in ${\gamma }_1+\dots {\gamma }_k=1.$

We will now present some interesting examples of the Carathéodory functions with the form (10). For $k=2$, Formula (10) with $\gamma \in [0,1]$ becomes

$$\begin{array}{cc}\hfill q_1\left(z\right)& =\gamma {\displaystyle \frac{1+{ϵ}_{1}z}{1-{ϵ}_{1}z}}+(1-\gamma ){\displaystyle \frac{1+{ϵ}_{2}z}{1-{ϵ}_{2}z}}\hfill \\ \hfill & =\gamma (1+2{ϵ}_{1}z+2{ϵ}_1^2{z}^2+\cdots )+(1-\gamma )(1+2{ϵ}_{2}z+2{ϵ}_2^2{z}^2+\cdots )\hfill \\ \hfill & =1+2\sum _{s=1}^{\infty }(\gamma {ϵ}_1^s+(1-\gamma ){ϵ}_2^s)z^s,z\in \Delta ,\hfill \end{array}$$

(11)

and $q_1\in \mathcal{P}$. If ${ϵ}_1=1$ and ${ϵ}_2=-1$, then (11) becomes

$$\begin{array}{cc}\hfill q_2\left(z\right)& =\gamma {\displaystyle \frac{1+z}{1-z}}+(1-\gamma ){\displaystyle \frac{1-z}{1+z}}\hfill \\ \hfill & =1+(4\gamma -2)z+2z^2+(4\gamma -2)z^3+\cdots \hfill \\ \hfill & =1+p_{1}z+p_2{z}^2+\cdots ,z\in \Delta ,\hfill \end{array}$$

(12)

and the function $q_2\in \mathcal{P}$. If we want the first coefficient $p_1$ to be fixed $p_1=p$, $p\in (-2,2)$, in (12), then we put $\gamma =(2+p)/4>0$, and so (12) becomes

$$\begin{array}{cc}\hfill q_3\left(z\right)& ={\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{p}{2}}\right){\displaystyle \frac{1+z}{1-z}}+{\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{p}{2}}\right){\displaystyle \frac{1-z}{1+z}}\hfill \\ \hfill & =1+pz+2z^2+p{z}^3+2z^4+\dots .\hfill \end{array}$$

(13)

If $\gamma =1/2$, $p\in (-2,2)$,

$${ϵ}_1={\displaystyle \frac{p+i\sqrt{4-p^2}}{2}},{ϵ}_2={\displaystyle \frac{p-i\sqrt{4-p^2}}{2}},|{ϵ}_1| = |{ϵ}_2| = 1,$$

then (11) becomes

$$\begin{array}{cc}\hfill q_4\left(z\right)& ={\displaystyle \frac{1}{2}}·{\displaystyle \frac{1+{ϵ}_{1}z}{1-{ϵ}_{1}z}}+{\displaystyle \frac{1}{2}}·{\displaystyle \frac{1+{ϵ}_{2}z}{1-{ϵ}_{2}z}}\hfill \\ \hfill & =1+\sum _{s=1}^{\infty }({ϵ}_1^s+{ϵ}_2^s)z^s\hfill \\ \hfill & ={\displaystyle \frac{1-z^2}{1-pz+z^2}}\hfill \\ \hfill & =1+pz+(p^2-2)z^2+(p^3-3p)z^3+\dots ,z\in \Delta ,\hfill \end{array}$$

(14)

and we have $q_4\in \mathcal{P}$. Note that (13) may be generalized, and the class

$${\mathcal{P}}_k=\left\{q\left(z\right)={\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{k}{2}}\right)p_1\left(z\right)+{\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{k}{2}}\right)p_2\left(z\right):p_1,p_2\in \mathcal{P}\right\}$$

was considered by Pinchuk [23] (see also [24]). Putting $q_j$ of the form (13) or (14) in (9) gives the following functions that are elements of the class ${\mathcal{C}}_0(F_{\alpha },p)$:

$$f_3\left(z\right)=z+{\displaystyle \frac{1}{2}}p{z}^2+{\displaystyle \frac{1}{3}}(\alpha +2)z^3+{\displaystyle \frac{1}{4}}(\alpha +1)p{z}^4+{\displaystyle \frac{1}{5}}({\alpha }^2+2\alpha +2)z^5+\dots$$

(15)

and

$$f_4\left(z\right)=z+{\displaystyle \frac{1}{2}}p{z}^2+{\displaystyle \frac{1}{3}}(\alpha -2+p^2)z^3+{\displaystyle \frac{1}{4}}(p\alpha -3p+p^3)z^4+\dots .$$

(16)

The following results are well-known coefficient properties in class $\mathcal{P}$.

Lemma 2.

If $q\in \mathcal{P}$ has the form (2), then

$$\left|p_2-{\displaystyle \frac{\mu }{2}}{p}_1^2\right| \le max\{2,2|\mu -1\left|\right\}=\left\{\begin{array}{cc}2,\hfill & 0 \le \mu \le 2,\hfill \\ 2|\mu -1|,\hfill & \mathit{elsewhere}.\hfill \end{array}\right.$$

Lemma 3

([7,16] (p. 104)). If $q\in \mathcal{P}$ has the form (2), then

$$|p_{2k-1}-p_{2k-2}| \le (4k-3)|2-p_1|,k=2,3,4,\dots$$

and

$$|p_{2k}-p_{2k-1}| \le (4k-1)|2-p_1|,k=2,3,4,\dots .$$

Some further coefficient results for the class $\mathcal{P}$ can be found in [25] or [26].

3. Main Results

Let us recall that Equality (8) gives several dependencies connecting the coefficients of the function $f\in {\mathcal{C}}_0\left(F_{\alpha }\right)$ and the function $q\in \mathcal{P}$:

$$2a_2=p_1,n{a}_n-\alpha (n-2)a_{n-2}=p_{n-1},n=3,4,5,\dots .$$

(17)

In particular cases, we obtain

$$2a_2=p_1,3a_3=\alpha +p_2,4a_4=p_1\alpha +p_3,5a_5={\alpha }^2+p_2\alpha +p_4.$$

(18)

It was shown in [15] that (17) results in

$$a_{2m}={\displaystyle \frac{1}{2m}}\sum _{k=1}^m{p}_{2k-1}{\alpha }^{m-k}={\displaystyle \frac{1}{2m}}\left\{p_{2m-1}+\sum _{k=1}^{m-1}{p}_{2k-1}{\alpha }^{m-k}\right\}$$

(19)

and

$$a_{2m-1}={\displaystyle \frac{1}{2m-1}}\sum _{k=1}^m{p}_{2k-2}{\alpha }^{m-k}={\displaystyle \frac{1}{2m-1}}\left\{p_{2m-2}+\sum _{k=1}^{m-1}{p}_{2k-2}{\alpha }^{m-k}\right\},$$

(20)

where $m\in \mathbb{N}$.

Consider the following functional, which is defined for $f\in \mathcal{A}$ of Form (1):

$$F_n=a_{n+1}-a_n.$$

One can prove that [15], for $n=2m-1,m\in \mathbb{N}$

$$\begin{array}{cc}\hfill F_{2m-1}& =a_{2m}-a_{2m-1}\hfill \\ \hfill & ={\displaystyle \frac{1}{2m}}\sum _{k=1}^m{p}_{2k-1}{\alpha }^{m-k}-{\displaystyle \frac{1}{2m-1}}\sum _{k=1}^m{p}_{2k-2}{\alpha }^{m-k}\hfill \\ \hfill & ={\alpha }^{m-1}\left({\displaystyle \frac{p_1}{2m}}-{\displaystyle \frac{p_0}{2m-1}}\right)+{\alpha }^{m-2}\left({\displaystyle \frac{p_3}{2m}}-{\displaystyle \frac{p_2}{2m-1}}\right)+\dots \hfill \\ \hfill & +\left({\displaystyle \frac{p_{2m-1}}{2m}}-{\displaystyle \frac{p_{2m-2}}{2m-1}}\right),\hfill \end{array}$$

(21)

and, for $n=2m,m\in \mathbb{N}$

$$\begin{array}{cc}\hfill F_{2m}& =a_{2m+1}-a_{2m}\hfill \\ \hfill & ={\displaystyle \frac{1}{2m+1}}\sum _{k=1}^{m+1}{p}_{2k-2}{\alpha }^{m+1-k}-{\displaystyle \frac{1}{2m}}\sum _{k=1}^m{p}_{2k-1}{\alpha }^{m-k}\hfill \\ \hfill & ={\displaystyle \frac{p_0}{2m+1}}{\alpha }^m+{\alpha }^{m-1}\left({\displaystyle \frac{p_2}{2m+1}}-{\displaystyle \frac{p_1}{2m}}\right)+{\alpha }^{m-2}\left({\displaystyle \frac{p_4}{2m+1}}-{\displaystyle \frac{p_3}{2m}}\right)+\dots \hfill \\ & +\left({\displaystyle \frac{p_{2m}}{2m+1}}-{\displaystyle \frac{p_{2m-1}}{2m}}\right).\hfill \end{array}$$

(22)

For a fixed $p\in [-2,2]$, the following sharp estimates of $|F_2|$ for ${\mathcal{C}}_0(F_{\alpha },p)$ were given.

Theorem 1

([15]). If $f\in {\mathcal{C}}_0(F_{\alpha },p)$, $p\in [-2,2]$, and if f has the form (1), then the following sharp inequality holds:

$$|F_2|=|a_3-a_2| \le \left\{\begin{array}{cc}{\displaystyle \frac{1}{6}}(4+2\alpha -3p)\hfill & \mathit{when}p\in [-2,p_{\ast }],\hfill \\ {\displaystyle \frac{1}{6}}(4-2\alpha +3p-2p^2)\hfill & \mathit{when}p\in [p_{\ast },2],\hfill \end{array}\right.$$

(23)

where $p_{\ast }={\displaystyle \frac{1}{2}}(3-\sqrt{9-8\alpha })$.

The equality in (23) holds for Function (15) for case $p\in [-2,p_{\ast }]$. If $p\in [p_{\ast },2]$, then the equality in (23) is achieved for Function (16).

Moreover, the following estimates of $|F_3|$ for the class ${\mathcal{R}}_p$ of bounded turning with the second coefficient fixed were provided.

Theorem 2

([15]). If $f\in {\mathcal{R}}_p,p\in [-2,2]$ then the following inequalities hold:

$$|F_3| \le \left\{\begin{array}{cc}{\displaystyle \frac{1}{12}}(2p^2-3p)\sqrt{{\displaystyle \frac{-8p^3+12p^2-24p+16}{3p^4-8p^3}}}\hfill & \mathit{when}p\in [-2,t_1),\hfill \\ {\displaystyle \frac{1}{12}}(3p^3-4p^2-9p+8)\hfill & \mathit{when}p\in [t_1,-{\displaystyle \frac{1}{3}}],\hfill \\ {\displaystyle \frac{1}{18}}(13-8p)\hfill & \mathit{when}p\in (-{\displaystyle \frac{1}{3}},0),\hfill \\ {\displaystyle \frac{1}{18}}\left(13-2p\right)\hfill & \mathit{when}p\in [0,{\displaystyle \frac{5}{3}}),\hfill \\ {\displaystyle \frac{1}{12}}(-3p^3+4p^2+9p-8)\hfill & \mathit{when}p\in [{\displaystyle \frac{5}{3}},2],\hfill \end{array}\right.$$

(24)

where $t_1$ is the only root of the polynomial $T\left(p\right)=12p^3-32p^2-4/3p+16$ in the interval $(-1,0)$.

Remark 1.

We will give here some supplementation to Theorem 2. Note that, for the function $f_4$, which is given in (16) with $\alpha =0,$ we have

$$F_3=a_4-a_3={\displaystyle \frac{1}{12}}(3p^3-4p^2-9p+8)={\displaystyle \frac{1}{12}}W\left(p\right).$$

After analyzing the sign of the polynomial W in the interval $[-2,2]$, we find that

$$|F_3|={\displaystyle \frac{1}{12}}(3p^3-4p^2-9p+8)$$

for $p\in [t_1,-{\displaystyle \frac{1}{3}}]$, where $t_1$ is determined in Theorem 2, while for $p\in [{\displaystyle \frac{5}{3}},2]$, we have

$$|F_3|={\displaystyle \frac{1}{12}}(-3p^3+4p^2+9p-8).$$

This shows the sharpness of Estimate (24) in the case when $p\in \left[t_1,-{\displaystyle \frac{1}{3}}\right]\cup \left[{\displaystyle \frac{5}{3}},2\right].$

In this section, we continue the search for the estimates of $|F_n|$ for $f\in {\mathcal{C}}_0(F_{\alpha },p)$. In addition, we provide estimates of the functional $|2m{a}_{2m}-(2m-1)a_{2m-1}|$ and bounds of the n-th coefficient in both considered classes ${\mathcal{C}}_0\left(F_{\alpha }\right)$ and ${\mathcal{C}}_0(F_{\alpha },p)$.

Theorem 3.

If $f\in {\mathcal{C}}_0(F_{\alpha },p)$, $p\in [-2,2]$, $\alpha \in [-1,1]$, and if f has the form (1), then

$$|F_3| = |a_4-a_3| \le \left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}+\left|{\displaystyle \frac{3}{16}}\alpha -{\displaystyle \frac{2}{3}}\right|+{\displaystyle \frac{3}{64}}\left|\alpha \right|{\left(p-{\displaystyle \frac{8}{3}}\right)}^2\mathit{when}\hfill & \alpha \in [-1,0],\hfill \\ {\displaystyle \frac{7}{6}}+{\displaystyle \frac{3}{64}}\left|\alpha \right|{\left(p-{\displaystyle \frac{8}{3}}\right)}^2\mathit{when}\hfill & \alpha \in [0,1].\hfill \end{array}\right.$$

(25)

Proof. 

From (21) or from (18), we have

$$\begin{array}{cc}\hfill |a_4-a_3|& = \left|a_4-{\displaystyle \frac{\alpha a_2}{2}}+{\displaystyle \frac{\alpha a_2}{2}}-a_3\right|\hfill \\ & \le \left|a_4-{\displaystyle \frac{\alpha a_2}{2}}\right|+\left|{\displaystyle \frac{\alpha a_2}{2}}-a_3\right|\hfill \\ & = \left|{\displaystyle \frac{p_3}{4}}\right|+\left|{\displaystyle \frac{\alpha a_2}{2}}-a_3\right|\hfill \\ & = \left|{\displaystyle \frac{p_3}{4}}\right|+\left|{\displaystyle \frac{p_2}{3}}+{\displaystyle \frac{\alpha }{3}}-{\displaystyle \frac{p\alpha }{4}}\right|\hfill \\ & = \left|{\displaystyle \frac{p_3}{4}}\right|+{\displaystyle \frac{1}{3}}\left|\left(p_2-{\displaystyle \frac{9\alpha p^2}{64}}\right)+\alpha \left({\displaystyle \frac{9p^2}{64}}-{\displaystyle \frac{3p}{4}}+1\right)\right|\hfill \\ & \le \left|{\displaystyle \frac{p_3}{4}}\right|+{\displaystyle \frac{1}{3}}\left|p_2-{\displaystyle \frac{9\alpha p^2}{64}}\right|+{\displaystyle \frac{3}{64}}\left|\alpha \right|{\left(p-{\displaystyle \frac{8}{3}}\right)}^2.\hfill \end{array}$$

It follows from the known estimates for the class of Carathéodory functions that $|{\displaystyle \frac{p_3}{4}}| \le {\displaystyle \frac{1}{2}}$. Moreover, using Lemma 2 with $\mu ={\displaystyle \frac{9}{32}}\alpha$, we obtain

$${\displaystyle \frac{1}{3}}\left|p_2-{\displaystyle \frac{9\alpha }{64}}{p}_1^2\right| \le \left\{\begin{array}{cc}\left|{\displaystyle \frac{3}{16}}\alpha -{\displaystyle \frac{2}{3}}\right|\hfill & \mathrm{when}\alpha \in [-1,0],\hfill \\ {\displaystyle \frac{2}{3}}\hfill & \mathrm{when}\alpha \in [0,1].\hfill \end{array}\right.$$

Hence, returning to the estimation of $|F_3|$, we obtain, in the case when $\alpha \in [-1,0]$, the following:

$$\begin{array}{cc}\hfill |F_3|& \le \left|{\displaystyle \frac{p_3}{4}}\right|+{\displaystyle \frac{1}{3}}\left|p_2-{\displaystyle \frac{9\alpha p^2}{64}}\right|+{\displaystyle \frac{1}{64}}\left|\alpha \right|{\left(p-{\displaystyle \frac{8}{3}}\right)}^2 \le {\displaystyle \frac{1}{2}}+\left|{\displaystyle \frac{3}{16}}\alpha -{\displaystyle \frac{2}{3}}\right|+{\displaystyle \frac{3}{64}}\left|\alpha \right|{\left(p-{\displaystyle \frac{8}{3}}\right)}^2,\hfill \end{array}$$

or, in the case when $\alpha \in [0,1]$, the following is obtained instead:

$$\begin{array}{cc}\hfill |F_3|& \le {\displaystyle \frac{1}{2}}+{\displaystyle \frac{2}{3}}+{\displaystyle \frac{3}{64}}\left|\alpha \right|{\left(p-{\displaystyle \frac{8}{3}}\right)}^2={\displaystyle \frac{7}{6}}+{\displaystyle \frac{3}{64}}\left|\alpha \right|{\left(p-{\displaystyle \frac{8}{3}}\right)}^2.\hfill \end{array}$$

Thus, the proof is completed. □

We may generally establish the upper bound for $|F_{2m-1}|$ and for $|F_{2m}|$ even in the case when p is a complex number, but the result does not seem very good. This is because, in the proof, we apply the triangle inequality of type $|x{p}_k+y{p}_m| \le |x{p}_k| + |y{p}_m|$ many times. But the equality $|x{p}_k+y{p}_m| = |x{p}_k| + |y{p}_m|$ holds for the case when $arg\left\{x{p}_k\right\}=arg\left\{y{p}_m\right\}$ only.

Theorem 4.

If $f\in {\mathcal{C}}_0(F_{\alpha },p)$, $p\in \mathbb{C}$, $\alpha \in [-1,1]$, and if f has the form (1), then

$$\begin{array}{cc}\hfill |F_{2m-1}|& = |a_{2m}-a_{2m-1}|\hfill \\ & \le {\displaystyle \frac{1}{2m}}\left[\left|p-{\displaystyle \frac{2m}{2m-1}}\right|{\left|\alpha \right|}^{m-1}+\sum _{k=2}^m\left((4k-3)|2-p|+{\displaystyle \frac{2}{2m-1}}\right){\left|\alpha \right|}^{m-k}\right].\hfill \end{array}$$

(26)

Proof. 

Directly from (21), we have

$$\begin{array}{cc}\hfill a_{2m}-a_{2m-1}& =\sum _{k=1}^m\left({\displaystyle \frac{1}{2m}}{p}_{2k-1}-{\displaystyle \frac{1}{2m-1}}{p}_{2k-2}\right){\alpha }^{m-k}\hfill \\ \hfill & =\sum _{k=1}^m\left({\displaystyle \frac{1}{2m}}{p}_{2k-1}-{\displaystyle \frac{1}{2m}}{p}_{2k-2}+{\displaystyle \frac{1}{2m}}{p}_{2k-2}-{\displaystyle \frac{1}{2m-1}}{p}_{2k-2}\right){\alpha }^{m-k}\hfill \\ \hfill & =\sum _{k=1}^m\left[{\displaystyle \frac{1}{2m}}(p_{2k-1}-p_{2k-2})+\left({\displaystyle \frac{1}{2m}}-{\displaystyle \frac{1}{2m-1}}\right)p_{2k-2}\right]{\alpha }^{m-k}\hfill \\ \hfill & =\sum _{k=1}^m\left[{\displaystyle \frac{1}{2m}}(p_{2k-1}-p_{2k-2})+{\displaystyle \frac{1}{2m(2m-1)}}{p}_{2k-2}\right]{\alpha }^{m-k}\hfill \\ \hfill & =\left[{\displaystyle \frac{p_1-1}{2m}}-{\displaystyle \frac{1}{2m(2m-1)}}\right]{\alpha }^{m-1}+\sum _{k=2}^m\left[{\displaystyle \frac{p_{2k-1}-p_{2k-2}}{2m}}+{\displaystyle \frac{p_{2k-2}}{2m(2m-1)}}\right]{\alpha }^{m-k}\hfill \\ \hfill & =\left[{\displaystyle \frac{p_1}{2m}}-{\displaystyle \frac{1}{2m-1}}\right]{\alpha }^{m-1}+\sum _{k=2}^m\left[{\displaystyle \frac{p_{2k-1}-p_{2k-2}}{2m}}+{\displaystyle \frac{p_{2k-2}}{2m(2m-1)}}\right]{\alpha }^{m-k}\hfill \\ \hfill & ={\displaystyle \frac{1}{2m}}\left\{\left[p_1-{\displaystyle \frac{2m}{2m-1}}\right]{\alpha }^{m-1}+\sum _{k=2}^m\left[p_{2k-1}-p_{2k-2}+{\displaystyle \frac{p_{2k-2}}{2m-1}}\right]{\alpha }^{m-k}\right\},\hfill \end{array}$$

and, consequently, using known estimates of coefficients in the class $\mathcal{P}$ and Lemma 3, we obtain

$$\begin{array}{cc}\hfill |a_{2m}-a_{2m-1}|& \le {\displaystyle \frac{1}{2m}}\left[\left|p-{\displaystyle \frac{2m}{2m-1}}\right|{\left|\alpha \right|}^{m-1}+\sum _{k=2}^m\left(|p_{2k-1}-p_{2k-2}|+{\displaystyle \frac{|p_{2k-2}|}{2m-1}}\right){\left|\alpha \right|}^{m-k}\right]\hfill \\ \hfill & \le {\displaystyle \frac{1}{2m}}\left[\left|p-{\displaystyle \frac{2m}{2m-1}}\right|{\left|\alpha \right|}^{m-1}+\sum _{k=2}^m\left((4k-3)|2-p|+{\displaystyle \frac{2}{2m-1}}\right){\left|\alpha \right|}^{m-k}\right],\hfill \end{array}$$

which completes the proof. □

Theorem 5.

If $f\in {\mathcal{C}}_0(F_{\alpha },p)$, $p\in \mathbb{C}$, $\alpha \in [-1,1]$, and if f has the form (1), then

$$|F_{2m}|=|a_{2m+1}-a_{2m}| \le {\displaystyle \frac{{\left|\alpha \right|}^n}{2m+1}}+\sum _{k=1}^m\left({\displaystyle \frac{(4k-1)|2-p|}{2m+1}}+{\displaystyle \frac{2}{2m(2m+1)}}\right){\left|\alpha \right|}^{m-k}.$$

Proof. 

From (22), we have

$$\begin{array}{cc}\hfill F_{2m}& =a_{2m+1}-a_{2m}\hfill \\ \hfill & ={\displaystyle \frac{{\alpha }^n}{2m+1}}+\sum _{k=1}^m\left({\displaystyle \frac{p_{2k}}{2m-1}}-{\displaystyle \frac{p_{2k-1}}{2m}}\right){\alpha }^{m-k}\hfill \\ \hfill & ={\displaystyle \frac{{\alpha }^n}{2m+1}}+\sum _{k=1}^m\left({\displaystyle \frac{p_{2k}-p_{2k-1}}{2m-1}}-{\displaystyle \frac{p_{2k-1}}{2m(2m+1)}}\right){\alpha }^{m-k}.\hfill \end{array}$$

Now, from the second inequality in Lemma 3, and from the known fact that $|p_{2k-1}| \le 2$, we obtain

$$\begin{array}{cc}\hfill |F_{2m}|& =|a_{2m+1}-a_{2m}|\hfill \\ \hfill & \le {\displaystyle \frac{{\left|\alpha \right|}^n}{2m+1}}+\sum _{k=1}^m\left({\displaystyle \frac{(4k-1)|2-p|}{2m+1}}+{\displaystyle \frac{2}{2m(2m+1)}}\right){\left|\alpha \right|}^{m-k},\hfill \end{array}$$

and the proof is completed. □

For $m=2$ or $m=3$, Inequality (26) gives

$$|F_3|=|a_4-a_3| \le {\displaystyle \frac{1}{4}}\left[\left|\alpha \right|\left|p-{\displaystyle \frac{4}{3}}\right|+5|2-p|+{\displaystyle \frac{2}{3}}\right]$$

(27)

and

$$|F_5|=|a_6-a_5| \le {\displaystyle \frac{1}{6}}\left[\left|p-{\displaystyle \frac{6}{5}}\right|{\left|\alpha \right|}^2+\left(5|2-p|+{\displaystyle \frac{2}{5}}\right)\left|\alpha \right|+9|2-p|+{\displaystyle \frac{2}{5}}\right].$$

In general, it is hard to compare (27), where p is complex, with (25), where p is real. If, for example, $\alpha =0$, then, from (25), we have $|F_3| \le 7/6$, while (27) gives

$$|F_3|=|a_4-a_3| \le {\displaystyle \frac{1}{4}}\left[5|2-p|+{\displaystyle \frac{2}{3}}\right].$$

Then, we have

$${\displaystyle \frac{1}{4}}\left[5|2-p|+{\displaystyle \frac{2}{3}}\right]<{\displaystyle \frac{7}{6}}\iff |2-p|<{\displaystyle \frac{4}{5}},$$

so, in this case, Estimation (27) is better than (25) when $|2-p|<{\displaystyle \frac{4}{5}}$.

If we consider $\left|2m{a}_{2m}-(2m-1)a_{2m-1}\right|$ instead of $\left|a_{2m}-a_{2m-1}\right|$, then we can establish the upper bound even for the case where p is a complex number. Before we present this result, we define the functional

$$G_n=(n+1)a_{n+1}-n{a}_n$$

and then recall a related result due to Zaprawa and Trąbka-Więcław, thereby giving estimates of $G_n$ for $f\in {\mathcal{C}}_0({\displaystyle \frac{z}{1-z^2}},p)$, which are only sharp for even n and for all positive integers n in the case when $p=-2.$

Theorem 6

([9]). If $\mathrm{Re}\left\{(1-z^2)f^{\prime }\left(z\right)\right\}>0,z\in \Delta$ and $f^{″}\left(0\right)=p,p\in [-2,2]$, then

$$|G_n| \le n+1-{\displaystyle \frac{1}{2}}np\mathit{if}\mathit{n}\mathit{is}\mathit{even}$$

and

$$|G_n| \le \left\{\begin{array}{cc}n+1-{\displaystyle \frac{1}{2}}np,\hfill & p\in [-2,0],\hfill \\ n+1-{\displaystyle \frac{1}{2}}(n-2)p,\hfill & p\in [0,2]\mathit{if}n\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(28)

For each $p\in [-2,2]$ and even n, the equality holds for function

$$f\left(z\right)={\displaystyle \frac{z+\frac{p}{2}{z}^2}{1-z^2}}=z+{\displaystyle \frac{p}{2}}{z}^2+z^3+{\displaystyle \frac{p}{2}}{z}^4+\dots .$$

Moreover, for $p=-2$ and all positive integers n, the equality holds for $f\left(z\right)=z/(1+z).$

Now, we present an estimate of $|G_n|$ for an odd positive integer n and complex p, which is sharp for $p=2.$

Theorem 7.

If $f\in {\mathcal{C}}_0(F_{\alpha },p)$, $\alpha \in [-1,1]$, $p\in \mathbb{C}$, and if f has the form (1), then, for all $m\in \mathbb{N}$, we have

$$\left|G_{2m-1}\right|=\left|2m{a}_{2m}-(2m-1)a_{2m-1}\right| \le {|p-1|\left|\alpha \right|}^{m-1}+|2-p|\sum _{k=2}^m(4k-3){\left|\alpha \right|}^{m-k}.$$

(29)

Proof. 

From (19) and (20), we have

$$2m{a}_{2m}-(2m-1)a_{2m-1}=\sum _{k=1}^m\left(p_{2k-1}-p_{2k-2}\right){\alpha }^{m-k},$$

where $p_k$ are the coefficients of the function $q\in \mathcal{P}$, which satisfies (9). Hence, using Lemma 2, we obtain

$$\begin{array}{cc}\hfill \left|G_{2m-1}\right|=\left|2m{a}_{2m}-(2m-1)a_{2m-1}\right|& \le |p-p_0{\left|\right|\alpha |}^{m-1}+\sum _{k=2}^m{(4k-3)|2-p|\left|\alpha \right|}^{m-k}\hfill \\ \hfill & ={|p-1|\left|\alpha \right|}^{m-1}+|2-p|\sum _{k=2}^m(4k-3){\left|\alpha \right|}^{m-k}.\hfill \end{array}$$

For $p=2$,

$$f\left(z\right)=z+z^2+{\displaystyle \frac{1}{3}}(\alpha +2)z^3+{\displaystyle \frac{1}{2}}(\alpha +1)z^4+\dots ,$$

(30)

that is, for the function $f_3$, which is given in (15) with $p=2$. □

In particular, for $m=2$, we have the following estimate:

$$\left|G_3\right|=|4a_4-3a_3| \le |p-1\left|\right|\alpha |+5(2-p),$$

and, for $m=3$, we obtain

$$\left|G_5\right|=|6a_6-5a_5{| \le |p-1\left|\right|\alpha |}^2+|2-p|\left(5\left|\alpha \right|+9\right).$$

Applying Theorem 7 with $p=2$, we can derive the following corollary.

Corollary 1.

If $f\in {\mathcal{C}}_0(F_{\alpha },2)$, $\alpha \in [-1,1]$, then, for all $m\in \mathbb{N}$, we have

$$\left|G_{2m-1}\right| \le {\left|\alpha \right|}^{m-1}.$$

(31)

The equality occurs for the function given in (30).

Remark 2.

Putting $\alpha =1$ in (31) gives the sharp result $\left|G_{2m-1}\right| \le 1$ for the class ${\mathcal{C}}_0({\displaystyle \frac{z}{1-z^2}},2)$, which coincides with that given in Theorem 6 for n odd and $p=2$. This shows that the result (28) is also sharp for $p=2$.

In the following theorems, we provide sharp estimates of the $n-$th coefficient for $f\in {\mathcal{C}}_0\left(F_{\alpha }\right)$.

Theorem 8.

If $f\in {\mathcal{C}}_0\left(F_{\alpha }\right)$, $\alpha \in [-1,1]$, and if f has the form (1), then

$$|a_{2m}| \le {\displaystyle \frac{1}{m}}\sum _{k=1}^m{\left|\alpha \right|}^{m-k}$$

(32)

and

$$|a_{2m-1}| \le {\displaystyle \frac{1}{2m-1}}{\left|\alpha \right|}^{m-1}+{\displaystyle \frac{2}{2m-1}}\sum _{k=2}^m{\left|\alpha \right|}^{m-k}.$$

(33)

The bounds (32) and (33) are sharp.

Proof. 

From (19), we have that the even coefficients $a_{2m}$ satisfy

$$\begin{array}{cc}\hfill |2m{a}_{2m}|& =\left|\sum _{k=1}^m{p}_{2k-1}{\alpha }^{m-k}\right| \le \sum _{k=1}^m|p_{2k-1}|{\left|\alpha \right|}^{m-k} \le 2\sum _{k=1}^m{\left|\alpha \right|}^{m-k}\hfill \end{array}$$

because $|p_s| \le 2,s=1,3,\dots$. This proves (32). For the odd coefficients $a_{2m-1}$, from (20), we have

$$\begin{array}{cc}\hfill |a_{2m-1}|& =\left|{\displaystyle \frac{1}{2m-1}}\sum _{k=1}^m{p}_{2k-2}{\alpha }^{m-k}\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2m-1}}\left[{\left|\alpha \right|}^{m-1}|p_0|+\sum _{k=2}^{m}2{\left|\alpha \right|}^{m-k}\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2m-1}}{\left|\alpha \right|}^{m-1}+{\displaystyle \frac{2}{2m-1}}\sum _{k=2}^m{\left|\alpha \right|}^{m-k}.\hfill \end{array}$$

As such, we obtain (33). Now, we will try to establish some functions $f\in {\mathcal{C}}_0\left(F_{\alpha }\right)$ that show the sharpness of the bounds (32) and (33). From (9), it is known that, if $q_0\in \mathcal{P}$, then there exists $f_0\in {\mathcal{C}}_0\left(F_{\alpha }\right)$ such that

$$(1-\alpha z^2)f_0^{\prime }\left(z\right)=q_0\left(z\right)={\displaystyle \frac{1+z}{1-z}}=1+2z+2z^2+2z^3+\dots ,z\in \Delta .$$

(34)

Then, from (19) and (20), the coefficients of $f_0\left(z\right)=z+a_2{z}^2+a_3{z}^3+\cdots$, where $f_0$ is described in (34), satisfy

$$2m{a}_{2m}=2\sum _{k=1}^m{\alpha }^{m-k}$$

and

$$(2m-1)a_{2m-1}={\alpha }^{m-1}+2\sum _{k=2}^m{\alpha }^{m-k}.$$

This shows that Estimations (32) and (33) are sharp in the case when $\alpha \in [0,1]$.

Note that the $f_0\in {\mathcal{C}}_0\left(F_{\alpha }\right)$ defined in (34) is of the form

$$\begin{array}{cc}\hfill f_0\left(z\right)& =z+z^2+{\displaystyle \frac{\alpha +2}{3}}{z}^3+{\displaystyle \frac{\alpha +1}{2}}{z}^4+\dots \hfill \\ \hfill & =z+z^2+\sum _{m=2}^{\infty }\left[{\displaystyle \frac{1}{2m-1}}\left({\alpha }^{m-1}+2\sum _{k=2}^m{\alpha }^{m-k}\right)z^{2m-1}+{\displaystyle \frac{1}{m}}\left(\sum _{k=1}^m{\alpha }^{m-k}\right)z^{2m}\right].\hfill \end{array}$$

Observe that the coefficients of $f_0$ are not strictly decreasing. If $\alpha =1$, then the above function $f_0$ reduces to

$${\displaystyle \frac{z}{1-z}}=z+z^2+z^3+\cdots .$$

If $\alpha \in [-1,0)$ and m is odd, then, from (19), we have

$$\begin{array}{cc}\hfill 2m{a}_{2m}& =\sum _{k=1}^m{p}_{2k-1}{\alpha }^{m-k}\hfill \\ \hfill & =p_1{\left|\alpha \right|}^{m-1}-p_3{\left|\alpha \right|}^{m-2}+p_5{\left|\alpha \right|}^{m-3}-p_7{\left|\alpha \right|}^{m-4}+\dots -p_{2m-3}\left|\alpha \right|+p_{2m-1}.\hfill \end{array}$$

If in the above equality, we apply the odd coefficients $p_{2k-1}$ of the Carathéodory’s function

$$q_0(-iz)={\displaystyle \frac{1-iz}{1+iz}}=1-2iz-2z^2+2i{z}^3+2z^4-2i{z}^5-2z^6+2i{z}^7+2z^8+\cdots ,$$

(35)

then, from (34), we can obtain a suitable function $f_0\in {\mathcal{C}}_0\left(F_{\alpha }\right)$ whose even coefficients $a_{2m}$ satisfy

$$\begin{array}{cc}\hfill 2im{a}_{2m}& ={2\left|\alpha \right|}^{m-1}+{2\left|\alpha \right|}^{m-2}+{2\left|\alpha \right|}^{m-3}+{2\left|\alpha \right|}^{m-4}+\cdots +2\left|\alpha \right|+1.\hfill \end{array}$$

This shows that equality occurs in (32) when m is odd. If $\alpha \in [-1,0)$ and m is even, then, from (19), we have

$$\begin{array}{cc}\hfill 2m{a}_{2m}& =\sum _{k=1}^m{p}_{2k-1}{\alpha }^{m-k}\hfill \\ \hfill & =-p_1{\left|\alpha \right|}^{m-1}+p_3{\left|\alpha \right|}^{m-2}-p_5{\left|\alpha \right|}^{m-3}+p_7{\left|\alpha \right|}^{m-4}-\dots -p_{2m-3}\left|\alpha \right|+p_{2m-1}.\hfill \end{array}$$

If, in the above equality, we apply the even coefficients $p_{2k-1}$ of the Carathéodory’s function (35), then

$$\begin{array}{cc}\hfill -2im{a}_{2m}& ={2\left|\alpha \right|}^{m-1}+{2\left|\alpha \right|}^{m-2}+{2\left|\alpha \right|}^{m-3}+{2\left|\alpha \right|}^{m-4}+\dots +2\left|\alpha \right|+1.\hfill \end{array}$$

This shows that we obtain the equality in (32) when m is even. To show the sharpness of (33), one must first consider $\alpha \in [-1,0)$ and an odd m. Then, from (20), we have

$$\begin{array}{cc}\hfill a_{2m-1}& ={\displaystyle \frac{1}{2m-1}}\sum _{k=1}^m{p}_{2k-2}{\alpha }^{m-k}\hfill \\ \hfill & ={\displaystyle \frac{1}{2m-1}}\left(p_0{\left|\alpha \right|}^{m-1}-p_2{\left|\alpha \right|}^{m-2}+p_4{\left|\alpha \right|}^{m-3}-p_6{\left|\alpha \right|}^{m-4}+\dots -p_{2m-4}\left|\alpha \right|+p_{2m-2}\right).\hfill \end{array}$$

If, in the above equality, we apply the even coefficients $p_{2k-2}$ of the Carathéodory’s function

$$q_0\left(iz\right)={\displaystyle \frac{1+iz}{1-iz}}=1+2iz-2z^2-2i{z}^3+2z^4+2i{z}^5-2z^6-2i{z}^7+2z^8+\dots ,$$

(36)

then we obtain

$$\begin{array}{cc}\hfill a_{2m-1}& ={\displaystyle \frac{1}{2m-1}}{\left|\alpha \right|}^{m-1}+{\displaystyle \frac{2}{2m-1}}\left({\left|\alpha \right|}^{m-2}+{\left|\alpha \right|}^{m-3}+{\left|\alpha \right|}^{m-4}+\dots +\left|\alpha \right|+1\right).\hfill \end{array}$$

This shows that we obtain the equality in (33) when m is odd. If $\alpha \in [-1,0)$, and m is even, then, from (20), we have

$$\begin{array}{cc}\hfill a_{2m-1}& ={\displaystyle \frac{1}{2m-1}}\sum _{k=1}^m{p}_{2k-2}{\alpha }^{m-k}\hfill \\ \hfill & ={\displaystyle \frac{1}{2m-1}}\left(-p_0{\left|\alpha \right|}^{m-1}+p_2{\left|\alpha \right|}^{m-2}-p_4{\left|\alpha \right|}^{m-3}+\dots +p_{2m-4}\left|\alpha \right|-p_{2m-2}\right).\hfill \end{array}$$

If, in the above equality, we apply the even coefficients $p_{2k-2}$ of the (36), then

$$\begin{array}{cc}\hfill -a_{2m-1}& ={\displaystyle \frac{1}{2m-1}}{\left|\alpha \right|}^{m-1}+{\displaystyle \frac{2}{2m-1}}\left({\left|\alpha \right|}^{m-2}+{\left|\alpha \right|}^{m-3}+{\left|\alpha \right|}^{m-4}+\dots +\left|\alpha \right|+1\right).\hfill \end{array}$$

This shows that we obtain the equality in (33) when m is even. □

If we consider class ${\mathcal{C}}_0(F_{\alpha },p)$ with a fixed real $a_2=p/2$, then Theorem 8 becomes the following result.

Theorem 9.

If $f\in {\mathcal{C}}_0(F_{\alpha },p)$, $p\in \mathbb{C}$, $\alpha \in [-1,1]$, and if f has the form (1), then

$$|a_{2m}| \le {\displaystyle \frac{1}{2m}}\left[{\left|p\right|\left|\alpha \right|}^{m-1}+2\sum _{k=2}^m{\left|\alpha \right|}^{m-k}\right]$$

(37)

and

$$|a_{2m-1}| \le {\displaystyle \frac{1}{2m-1}}{\left|\alpha \right|}^{m-1}+{\displaystyle \frac{2}{2m-1}}\sum _{k=2}^m{\left|\alpha \right|}^{m-k}.$$

(38)

Proof. 

From (19), we have

$$\begin{array}{cc}\hfill |a_{2m}|& ={\displaystyle \frac{1}{2m}}\left|\sum _{k=1}^m{p}_{2k-1}{\alpha }^{m-k}\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2m}}\sum _{k=1}^m|p_{2k-1}{\left|\right|\alpha |}^{m-k}\hfill \\ \hfill & ={\displaystyle \frac{1}{2m}}\left[{\left|p\right|\left|\alpha \right|}^{m-1}+2\sum _{k=1}^m{\left|\alpha \right|}^{m-k}\right],\hfill \end{array}$$

which gives (37). For the odd coefficients $a_{2m-1}$, we apply (20), to have

$$\begin{array}{cc}\hfill |a_{2m-1}|& =\left|{\displaystyle \frac{1}{2m-1}}\sum _{k=1}^m{p}_{2k-2}{\alpha }^{m-k}\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2m-1}}\left[{\left|\alpha \right|}^{m-1}|p_0|+\sum _{k=2}^{m}2{\left|\alpha \right|}^{m-k}\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2m-1}}{\left|\alpha \right|}^{m-1}+{\displaystyle \frac{2}{2m-1}}\sum _{k=2}^m{\left|\alpha \right|}^{m-k}.\hfill \end{array}$$

This gives (38). □

An open problem is whether the bounds in (37) and (38) are sharp. The extremal functions that were applied in the proof of Theorem 8 are not in the class ${\mathcal{C}}_0(F_{\alpha },p)$ for all $p\in (-2,2)$ because, in each case of these functions, the second coefficient $a_2=p_1/2$ is not in $(-1,1)$.

4. Conclusions

We have provided the estimates of the functionals $|a_{n+1}-a_n|$, $|2n{a}_{2n}-(2n-1)a_{2n-1}|$ and the sharp estimates of the $n-$th coefficient for the class ${\mathcal{C}}_0\left(F_{\alpha }\right)$. These results were obtained by relating this class to the Carathéodory class of functions with positive real part and using Lemmas 2 and 3. The search for such estimates is one of the classical problems in geometric function theory. Another classical problem is the Fekete–Szeg$\ddot{\mathrm{o}}$ problem, which is related to the search for maximum value of the coefficient functional

$${\mathrm{\Phi }}_{\mu }\left(f\right)=|a_3-\mu a_2^2|,\mu \in \mathbb{C}$$

when f varies over the class $\mathcal{S}$. In 1933, Fekete and Szeg$\ddot{\mathrm{o}}$ [27] proved that

$$\underset{f\in \mathcal{S}}{max}{\Phi }_{\mu }\left(f\right)=\left\{\begin{array}{cc}1+2e^{-2\mu /\left(\right)1-\mu }\hfill & \mathrm{for}0 \le \mu <1,\hfill \\ 1\hfill & \mathrm{for}\mu =1.\hfill \end{array}\right.$$

The sharp bound of ${\Phi }_{\mu }\left(f\right)$ for the real $\mu$ for the class of close-to-convex functions was given in 1987 by Koepf [28]. Many mathematicians have been interested in finding the maximum of the Fekete–Szeg$\ddot{\mathrm{o}}$ functional ${\Phi }_{\mu }\left(f\right)$ in various subclasses of univalent functions (see [29,30,31,32]). In [33], the following sharp result for $\mu \in [0,1]$ for the class $\mathcal{R}$ of bounded turning was given

$$\underset{f\in \mathcal{R}}{max}{\Phi }_{\mu }\left(f\right)={\displaystyle \frac{2}{3}}.$$

As such, we know the maximum value of the functional ${\Phi }_{\mu }\left(f\right)$ for the class $\mathcal{R}={\mathcal{C}}_0\left(F_0\right).$ Therefore, it seems to be a natural direction for further research to consider the Fekete–Szeg$\ddot{\mathrm{o}}$ problem for the class ${\mathcal{C}}_0\left(F_{\alpha }\right),\alpha \in [-1,1].$ It may also be interesting to examine estimates of the Hankel determinants for this class.