ON CERTAIN CLOSE-TO-CONVEX FUNCTIONS

Abstract

Let $\mathcal {K}_u$ denote the class of all analytic functions f in the unit disk $\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}$, normalised by $f(0)=f'(0)-1=0$ and satisfying $|zf'(z)/g(z)-1|<1$ in $\mathbb {D}$ for some starlike function g. Allu, Sokól and Thomas [‘On a close-to-convex analogue of certain starlike functions’, Bull. Aust. Math. Soc. 108 (2020), 268–281] obtained a partial solution for the Fekete–Szegö problem and initial coefficient estimates for functions in $\mathcal {K}_u$, and posed a conjecture in this regard. We prove this conjecture regarding the sharp estimates of coefficients and solve the Fekete–Szegö problem completely for functions in the class $\mathcal {K}_u$.

1 Introduction

Let $\mathcal {H}$ be the class of all analytic functions in the unit disk $\mathbb {D}:=\{z\in \mathbb {C}:\, |z|<1\}$ . Let $\mathcal {B}$ be the subclass of $\mathcal {H}$ consisting of all functions f in $\mathcal {H}$ with $|f(z)|<1$ for all $z\in \mathbb {D}$ , $\mathcal {B}_0$ be the subclass of $\mathcal {B}$ with $f(0)=0$ and $\mathcal {A}$ be the subclass of $\mathcal {H}$ consisting of all functions f normalised by $f(0)=f'(0)-1=0$ with the Taylor series expansion

(1.1) $$ \begin{align} f(z)= z+\sum_{n=2}^{\infty}a_n z^n. \end{align} $$

Further, let $\mathcal {S}$ be the subclass of $\mathcal {A}$ that are univalent (that is, one-to-one) in $\mathbb {D}$ . A function $f\in \mathcal {A}$ is called starlike (respectively, convex) if $f(\mathbb {D})$ is a starlike domain (respectively, a convex domain) with respect to the origin. The set of all starlike functions and convex functions in $\mathcal {S}$ are denoted by $\mathcal {S}^*$ and $\mathcal {C}$ , respectively. It is well known that a function f in $\mathcal {A}$ is starlike (respectively, convex) if and only if $\mathrm {Re\,} zf'(z)/f(z)>0$ (respectively, $\mathrm {Re\,} (1+zf"(z)/f'(z))>0$ ) for $z\in \mathbb {D}$ . For further information about these classes, we refer to [5, 7].

A function $f\in \mathcal {A}$ is said to be close-to-convex if the complement of the image-domain $f(\mathbb {D})$ in $\mathbb {C}$ is the union of rays that are disjoint (except that the origin of one ray may lie on another one of the rays) and the class of all close-to-convex functions is denoted by $\mathcal {K}$ . This class was introduced by Kaplan [10]. A function $f\in \mathcal {A}$ is close-to-convex if and only if there exists a starlike function $g\in \mathcal {S}^*$ and a real number $\alpha \in (-\pi /2,\pi /2)$ such that (see [5, 10])

$$ \begin{align*} \mathrm{Re\,} \bigg(e^{i\alpha}\frac{zf'(z)}{g(z)}\bigg)>0,\quad z\in\mathbb{D}. \end{align*} $$

In 1968, Singh [16] introduced and studied the class $\mathcal {S}_u^*$ consisting of functions f in $\mathcal {A}$ such that

$$ \begin{align*} \bigg|\frac{zf'(z)}{f(z)}-1\bigg|<1\quad \text{for}~z\in\mathbb{D}. \end{align*} $$

It is easy to see that every function in $\mathcal {S}_u^*$ also belongs to $\mathcal {S}^*$ . Singh [16] obtained the distortion theorem, coefficient estimate and radius of convexity for the class $\mathcal {S}_u^*$ . Recently, Allu $et~ al.$ [1] introduced a close-to-convex analogue of the class $\mathcal {S}_u^*$ denoted by $\mathcal {K}_u$ . A function f in $\mathcal {A}$ belongs to $\mathcal {K}_u$ if there exists a starlike function $g\in \mathcal {S}^*$ such that

$$ \begin{align*} \bigg|\frac{zf'(z)}{g(z)}-1\bigg|<1\quad \text{for } z\in\mathbb{D}. \end{align*} $$

Clearly, every function in $\mathcal {K}_u$ is close-to-convex.

It is well known that if $f\in \mathcal {S}$ is of the form (1.1), then $|a_n|\le n$ for all $n\geq 2$ , and equality holds for the rotations of the Koebe function $k(z)=z/(1- z)^2$ . Singh [16] proved that if $f\in \mathcal {S}_u^*$ , then $|a_n|\le 1/(n-1)$ for all $~n\geq 2$ , and this inequality is sharp. In 2020, Allu $et~al.$ [1] studied coefficient bounds for the functions $f(z)$ of the form (1.1) in the class $\mathcal {K}_{u}$ and obtained the sharp bounds $|a_2|\le 3/2$ and $|a_3|\leq 5/3$ and proposed a conjecture that $|a_n|\le (2n-1)/n$ for $n\ge 4$ .

The Fekete–Szegö problem is to find the maximum value of the coefficient functional

$$ \begin{align*} \Phi_\mu(f)=|a_3-\mu a_2^2|, \quad \mu\in\mathbb{C}, \end{align*} $$

when f of the form (1.1) varies over a class of functions $\mathcal {F}$ . In 1933, Fekete–Szegö [6] used the Löwner differential method to prove that

$$ \begin{align*} \max\limits_{f\in\mathcal{S}} \Phi_\mu(f)= \begin{cases} 1+2e^{-{2\mu}/{(1-\mu)}} &\text{for }0\leq\mu<1, \\ 1 &\text{for } \mu=1. \end{cases} \end{align*} $$

In 1987, Koepf [12] obtained the sharp bound of $\Phi _\mu (f)$ for any $\mu \in \mathbb {R}$ for the class $\mathcal {K}$ :

$$ \begin{align*} \max\limits_{f\in\mathcal{K}} \Phi_\mu(f)=\begin{cases} |3-4\mu| &\text{if } \mu\in\bigg(-\infty,\dfrac{1}{3}\bigg]\cup[1,\infty),\\[6pt] \dfrac{1}{3}+\dfrac{4}{9\mu} &\text{if } \mu\in\bigg[\dfrac{1}{3},\dfrac{2}{3}\bigg],\\[6pt] 1 &\text{if } \mu\in\bigg[\dfrac{2}{3},1\bigg]. \end{cases} \end{align*} $$

The Fekete–Szegö problem has been studied for different subclasses of $\mathcal {S}$ (see [9, 13–15, 17]). Allu $et~ al.$ [1] considered the class $\mathcal {K}_{u}$ and obtained an estimate of the Fekete–Szegö functional $|a_3-\mu a_2^2|$ with $\mu \in \mathbb {R}$ . However, they were only able to show sharpness when $\mu \leq 0,~2/3\leq \mu \leq 1~\text {and}~\mu \geq 10/9$ .

Let $\mathcal {LU}$ denote the subclass of $\mathcal {H}$ consisting of all locally univalent functions in $\mathbb {D}$ , that is, $\mathcal {LU}:=\{f\in \mathcal {H}:f'(z)\ne 0\text { for all }z\in \mathbb {D}\}$ . For a locally univalent function $f\in \mathcal {LU}$ , the pre-Schwarzian derivative is defined by

$$ \begin{align*}P_f(z)=\frac{f"(z)}{f'(z)},\end{align*} $$

and the pre-Schwarzian norm (the hyperbolic sup-norm) is defined by

$$ \begin{align*} \|P_f\|=\sup\limits_{z\in\mathbb{D}}(1-|z|^2)|P_f(z)|. \end{align*} $$

This norm has significant meaning in the theory of Teichmüller spaces. For a univalent function f, it is well known that $\|P_f\|\leq 6$ and the estimate is sharp. However, if $\|P_f\|\leq 1$ , then f is univalent in $\mathbb {D}$ (see [2, 3]). In 1976, Yamashita [18] proved that $\|P_f \|$ is finite if and only if f is uniformly locally univalent in $\mathbb {D}$ . Moreover, if $\|P_f\|<2$ , then f is bounded in $\mathbb {D}$ (see [11]). We will obtain results related to the pre-Schwarzian norm for functions $f\in \mathcal {K}_u$ .

We first prove the conjecture $|a_n|\le (2n-1)/n$ for $n\ge 2$ for functions in $\mathcal {K}_u$ as proposed by Allu $et~ al.$ [1]. We next obtain the sharp estimate of the Fekete–Szegö functional $\Phi _\mu (f)$ for the class $\mathcal {K}_u$ for any $\mu \in \mathbb {R}$ . Finally, we obtain estimates of the pre-Schwarzian norm for functions in $\mathcal {K}_u$ .

2 Main results

Before stating our main results, we will discuss some preliminaries which will help us to prove our results. The first lemma is part of a result proved by Choi $et~al.$ [4].

Lemma 2.1. For $A, B\in \mathbb {C}$ and $K, L, M\in \mathbb {R}$ , let

$$ \begin{align*} \Omega_(A, B, K, L, M)=\max_{\substack{|u_1\leq 1\\ |v_1|\leq 1}}(|A|(1-|u_1|^2)+|B|(1-|v_1|^2)+|Ku_1^2+Lv_1^2+2Mu_1v_1|). \end{align*} $$

Further consider the following four conditions involving $A, B, K, L, M$ :

1. (A1) $|A|\geq \max \bigg \{|K|\sqrt {1-\dfrac {M^2}{KL}},|M|-|K|\bigg \}$ ;

2. (A2) $|K|+|M|\leq |A| <|K|\sqrt {1-\dfrac {M^2}{KL}}$ ;

3. (B1) $|B|\geq \max \bigg \{|L|\sqrt {1-\dfrac {M^2}{KL}},|M|-|L|\bigg \}$ ;

4. (B2) $|L|+|M|\leq |B| <|L|\sqrt {1-\dfrac {M^2}{KL}}$ .

If $KL\geq 0$ and $D=(|K|-|A|)(|L|-|B|)-M^2$ , then

$$ \begin{align*} \Omega(A, B, K, L, M)= \begin{cases} |A|+|L|-\dfrac{M^2}{|K|-|L|} &\text{if } |A|> |M|+|K|~\text{and}~D<0,\\ |B|+|K|-\dfrac{M^2}{|L|-|B|} &\text{if } |B|> |M|+|L|~\text{and}~D<0,\\ |K|+2|M|+|L| &\text{otherwise}. \end{cases} \end{align*} $$

If $KL<0$ , then $\Omega (A, B, K, L, M)=|A|+|B|+\max \{0,R\},$ where

$$ \begin{align*} R=\begin{cases} |K|-|A|+\dfrac{M^2}{|B|+|L|}, &\text{when (B1) holds but (A1) and (A2) do not hold}, \\ |L|-|B|+\dfrac{M^2}{|A|+|K|} , &\text{when (A1) holds but (B1) and (B2) do not hold}. \end{cases} \end{align*} $$

For two functions f and g in $\mathcal {H}$ , we say that $f(z)$ is majorised by $g(z)$ if $|f(z)|\leq |g(z)|$ for all $z\in \mathbb {D}$ or equivalently, if there exists $\omega \in \mathcal {B}$ such that ${f(z)=\omega (z)g(z)}$ . Let $f(z)=\sum _{n=0}^\infty a_nz^n$ and $F(z)=\sum _{n=0}^\infty A_nz^n$ be two power series convergent in some disk $E_R=\{z:|z|<R,~R>0\}$ . We say that $f(z)$ is dominated by $F(z)$ and we write $f(z)\ll F(z)$ if for any integer $n\geq 0$ , $|a_n|\leq |A_n|.$

Lemma 2.2 [8, Theorem 6.7].

If $f(z)=\sum _{n=1}^\infty a_nz^n$ , $z\in \mathbb {D}$ , is majorised by g and $g\in \mathcal {S}^*$ , then $|a_n|\le n$ for all $n\ge 1$ , that is, $f(z)\ll k(z)$ , where $k(z)=z/(1-z)^2$ is the Koebe function.

Our first result confirms the conjecture of Allu $et~al.$ in [1].

Theorem 2.3. Let $f\in \mathcal {K}_{u}$ be of the form (1.1). Then,

$$ \begin{align*}|a_n|\leq {\frac{2n-1}{n}}\quad\text{for all } n\geq 2.\end{align*} $$

Moreover, the estimate is sharp.

Proof. Let $f\in \mathcal {K}_{u}$ be of the form (1.1). Then there exists a starlike function $g\in \mathcal {S}^*$ such that

$$ \begin{align*}\bigg|\frac{zf'(z)}{g(z)}-1\bigg|<1.\end{align*} $$

Further, there exists a function $\omega (z)\in \mathcal {B}_{0}$ such that

$$ \begin{align*}zf'(z)=g(z)(1+\omega(z)),\end{align*} $$

that is,

(2.1) $$ \begin{align} zf'(z)=g(z)+zg(z)\omega_1(z) \end{align} $$

for some $\omega _1(z)\in \mathcal {B}$ . Since, $g(z)\omega _1(z)$ is majorised by $g(z)$ and $g\in \mathcal {S}^*$ , by Lemma 2.2, the function $g(z)\omega _1(z)$ is dominated by $k(z)$ , that is, $g(z)\omega _1(z)\ll k(z)$ . Thus, from (2.1),

$$ \begin{align*}zf'(z)\ll k(z)+zk(z),\end{align*} $$

and consequently,

$$ \begin{align*}|a_n|\leq \frac{2n-1}{n}.\end{align*} $$

The estimate is sharp for the function $f_1\in \mathcal {K}_{u}$ given by

$$ \begin{align*} f_1(z)=\frac{2z}{1-z}+\log(1-z).\\[-39pt] \end{align*} $$

For functions in $\mathcal {K}_{u}$ , Allu $et~ al.$ [1] obtained an estimate of the Fekete–Szegö functional $|a_3-\mu a_2^2|$ with $\mu \in \mathbb {R}$ . The result is sharp only when $\mu \leq 0,~2/3\leq \mu \leq ~1~\text {and}~\mu \geq 10/9$ . In the next theorem, we will give the sharp bounds of $|a_3-\mu a_2^2|$ for all values of $\mu \in \mathbb {R}$ . Our proof is completely different from that in [1]. Our main tool to get the sharp bound is Lemma 2.1.

Theorem 2.4. Let $f\in \mathcal {K}_{u}$ be given by (1.1). Then for every $\mu \in \mathbb {R}$ ,

$$ \begin{align*} |a_3-\mu a_2^2|\leq \begin{cases} \dfrac{5}{3}-\dfrac{9}{4}\mu &\text{if}\ \mu\leq0,\\[2mm] \dfrac{4(5-3\mu)}{3(4+3\mu)} &\text{if}\ 0\leq \mu \leq \dfrac{2}{3},\\[3mm] \dfrac{2}{3} &\text{if}\ \dfrac{2}{3}\leq \mu \leq 1,\\[1.5mm] \dfrac{3\mu-5}{3(3\mu-4)} &\text{if}\ 1\leq \mu \leq \dfrac{10}{9},\\[3mm] \dfrac{9}{4}\mu-\dfrac{5}{3} &\text{if}\ \mu\geq \dfrac{10}{9}. \end{cases} \end{align*} $$

Moreover, all the inequalities are sharp.

Proof. Let $f\in \mathcal {K}_{u}$ be of the form (1.1). Then there exists a starlike function $g(z)=z+\sum _{n=2}^{\infty }b_nz^n$ in $\mathcal {S}^* $ such that

$$ \begin{align*} \bigg|\frac{zf'(z)}{g(z)}-1\bigg|<1. \end{align*} $$

Thus, there exists $\omega (z)=\sum _{n=1}^{\infty }c_nz^n$ in $\mathcal {B}_0$ such that

(2.2) $$ \begin{align} f'(z)=\frac{g(z)}{z}(1+\omega(z)). \end{align} $$

From (2.2), comparing the coefficients of $z^2$ and $z^3$ on both sides,

(2.3) $$ \begin{align} a_2=\frac{b_2}{2}+\frac{c_1}{2}\quad\text{and}\quad a_3=\frac{c_2}{3}+\frac{b_3}{3}+\frac{1}{3}b_2c_1. \end{align} $$

Since $g\in \mathcal {S}^*$ , it follows that there exists another $\rho \in \mathcal {B}_0$ of the form $\rho (z)=\sum _{n=1}^{\infty }d_nz^n$ such that

(2.4) $$ \begin{align} \frac{zg'(z)}{g(z)}= \frac{1+ \rho(z)}{1- \rho(z)}. \end{align} $$

On comparing the coefficients of $z^2$ and $z^3$ on both sides,

(2.5) $$ \begin{align} b_2=2d_1\quad\text{and}\quad b_3=d_2+3d_1^2. \end{align} $$

From (2.3) and (2.5),

$$ \begin{align*} a_2=d_1+\frac{c_1}{2}\quad\text{and}\quad a_3=\frac{c_2}{3}+\frac{d_2}{3}+d_1^2+\frac{2}{3}d_1c_1. \end{align*} $$

Therefore, for any $\mu \in \mathbb {R}$ ,

$$ \begin{align*} a_3-\mu a_2^2=Ac_2+Bd_2+Kc_1^2+Ld_1^2+2Mc_1d_1, \end{align*} $$

where

$$ \begin{align*}A=\frac{1}{3}, \quad B=\frac{1}{3}, \quad K=-\frac{\mu}{4}, \quad M=\frac{2-3\mu}{6}, \quad L=1-\mu.\end{align*} $$

Thus,

$$ \begin{align*} |a_3-\mu a_2^2|&\leq |A\|c_2|+|B\|d_2|+|Kc_1^2+Ld_1^2+2Mc_1d_1|\\ &\leq |A|(1-|c_1|^2)+|B|(1-|d_1|^2)+|Kc_1^2+Ld_1^2+2Mc_1d_1|.\nonumber \end{align*} $$

Now, we have to find the maximum value of $|a_3-\mu a_2^2|$ when $|c_1|\leq 1,~ |d_1|\leq 1$ . To do this, we will use Lemma 2.1 and consider the following five cases.

Case 1: Let $\mu \leq 0$ . A simple calculation shows that

$$ \begin{align*}KL=-\frac{\mu(1-\mu)}{4}\geq0, \quad D=-\frac{2-3\mu}{6}<0, \quad |A|\le |M|+|K|, \quad |B|\le |M|+|L|.\end{align*} $$

Therefore, from Lemma 2.1,

$$ \begin{align*}|a_3-\mu a_2^2|\leq|K|+2|M|+|L|=\tfrac{5}{3}-\tfrac{9}{4}\mu.\end{align*} $$

The inequality is sharp and the equality holds for the function $f\in \mathcal {K}_{u}$ given by (2.2) and (2.4) with $\omega (z)=z$ and $\rho (z)=z$ , that is,

$$ \begin{align*}f(z)=\frac{2z}{1-z}+\log(1-z)=z+\frac{3}{2}z^2+\frac{5}{3}z^3+\cdots.\end{align*} $$

Case 2: Let $0\leq \mu \leq 2/3$ . A simple calculation shows that

$$ \begin{align*}KL=-\frac{\mu(1-\mu)}{4}<0.\end{align*} $$

Thus, from Lemma 2.1,

(2.6) $$ \begin{align} |a_3-\mu a_2^2|\leq|A|+|B|+\max\{0,R\}, \end{align} $$

where R can be obtained from Lemma 2.1. For $0\leq \mu \leq \tfrac 23$ ,

$$ \begin{align*} |M|-|K|=\frac{4-9\mu}{12}\le \frac{1}{3}=|A| \end{align*} $$

and

$$ \begin{align*} |K|\sqrt{1-\dfrac{M^2}{KL}}\leq|A|&\iff\dfrac{\mu}{4}\sqrt{1+\dfrac{(2-3\mu)^2}{9\mu(1-\mu)}} \le\frac{1}{3}\\ &\iff 3\mu^2-20\mu+16\geq0, \end{align*} $$

which is true for all $\mu \in [0,~2/3]$ . Thus, the condition (A1) of Lemma 2.1 is satisfied.

Again, for $0\leq \mu \leq 2/3$ ,

$$ \begin{align*} |M|-|L|=\frac{3\mu-4}{6}\leq -\frac{1}{3}\le |B| \end{align*} $$

and

$$ \begin{align*} |L|\sqrt{1-\dfrac{M^2}{KL}}\leq|B|&\iff(1-\mu)\sqrt{1+\dfrac{(2-3\mu)^2}{9\mu(1-\mu)}}\leq\frac{1}{3}\\ &\iff 3\mu^2-8\mu+4\leq 0, \end{align*} $$

which is not true for any $\mu \in [0,~2/3]$ . Thus, the condition (B1) of Lemma 2.1 is not satisfied. Further, for $0\leq \mu \leq 2/3$ ,

$$ \begin{align*}|L|+|M|=\frac{4-3\mu}{4}\ge \frac{1}{2}\ge |B|\end{align*} $$

and so, the condition (B2) of Lemma 2.1 is not satisfied.

Therefore, by Lemma 2.1,

$$ \begin{align*}R=|L|-|B|+\dfrac{M^2}{|A|+|K|}=\dfrac{2}{3}-\mu+\dfrac{(2-3\mu)^2}{4(4+3\mu)}\end{align*} $$

and consequently, from (2.6),

$$ \begin{align*} |a_3-\mu a_2^2|\leq\frac{4(5-3\mu)}{3(4+3\mu)}. \end{align*} $$

The inequality is sharp and the equality holds for the function $f\in \mathcal {K}_{u}$ given by (2.2) and (2.4) with

$$ \begin{align*} \omega(z)=\frac{az(z+\bar{a}v_1)}{1+a\bar{v_1}z}\quad \text{and}\quad \rho(z)=z, \end{align*} $$

where

$$ \begin{align*} v_1=\frac{2(2-3\mu)}{4+3\mu}\quad\text{and}\quad a=\frac{v_2}{1-v_1^2}\quad\text{with}\quad v_1^2+v_2=1,\end{align*} $$

that is,

$$ \begin{align*}f(z)=\int_0^z\frac{1+(a\bar{v_1}+v_1)t+at^2}{(1-t)^2(1+a\bar{v_1}t)}\,dt=z+\frac{6}{4+3\mu}z^2+\frac{80+120\mu-36\mu^2}{3(4+3\mu)^2}z^3+\cdots.\end{align*} $$

Case 3: Let $2/3\leq \mu \leq 1$ . It is easy to show that $KL= -\tfrac 14\mu (1-\mu )<0$ . So, from Lemma 2.1,

(2.7) $$ \begin{align} |a_3-\mu a_2^2|\leq|A|+|B|+\max\{0,R\}, \end{align} $$

where R can be obtained from Lemma 2.1. Proceeding as in Case 2, we can verify that the condition (A1) holds but (B1) and (B2) of Lemma 2.1 do not hold. Therefore,

$$ \begin{align*}R=|K|-|A|+\dfrac{M^2}{|B|+|L|}=\frac{\mu-1}{4-3\mu}\leq0\quad\text{for}~ \frac{2}{3}\leq\mu\leq 1\end{align*} $$

and consequently, from (2.7),

$$ \begin{align*}|a_3-\mu a_2^2|\leq \tfrac{1}{3}+\tfrac{1}{3}=\tfrac{2}{3}.\end{align*} $$

The inequality is sharp and the equality holds for the function $f\in \mathcal {K}_{u}$ given by (2.2) and (2.4) with $\omega (z)=z^2$ and $\rho (z)=z^2,$ that is,

$$ \begin{align*}f(z)=\log\frac{1+z}{1-z}-z=z+\frac{2}{3}z^3+\cdots.\end{align*} $$

Case 4: Let $1\leq \mu \leq 10/9$ . A simple calculation shows that

$$ \begin{align*}KL=-\frac{\mu(1-\mu)}{4}\geq0, \quad D=-\frac{1-\mu}{3}<0 \quad \text{and}\quad |B|>|M|+|L|.\end{align*} $$

Thus, from Lemma 2.1,

$$ \begin{align*}|a_3-\mu a_2^2|\leq|B|+|K|-\frac{M^2}{|L|-|B|}=\frac{5-3\mu}{3(4-3\mu)}.\end{align*} $$

The inequality is sharp and the equality holds for the function $f\in \mathcal {K}_{u}$ given by (2.2) and (2.4) with

$$ \begin{align*}\omega(z)=z \quad \text{and}\quad \rho(z)=\frac{az(z+\bar{a}v_1)}{1+a\bar{v_1}z},\end{align*} $$

where

$$ \begin{align*} v_1=\frac{3\mu-2}{8-6\mu}\quad \text{and} \quad a=-\frac{v_2}{1-v_1^2}\quad\text{with}\quad v_1^2+v_2=1,\end{align*} $$

that is,

$$ \begin{align*} \frac{g(z)}{z}=\exp\bigg( \int_0^z\frac{2(v_1+at)}{1+a\bar{v_1}t-v_1t-at^2}\,dt\bigg)=1+2v_1z+(4v_1^2-1)z^2+\cdots\end{align*} $$

and

$$ \begin{align*}f(z)=\int_0^z\frac{g(t)}{t}(1+t)\,dt=z+\frac{1}{4-3\mu}z^2+\frac{30\mu-20-9\mu^2}{3(4-3\mu)^2}z^3+\cdots.\end{align*} $$

Case 5: Let $\mu \geq 10/9$ . A simple calculation shows that

$$ \begin{align*}KL=-\frac{\mu(1-\mu)}{4}\geq0, \quad D=-\frac{1-\mu}{3}<0, \quad |A|\le |M|+|K|, \quad |B|\le |M|+|L|.\end{align*} $$

Thus, from Lemma 2.1,

$$ \begin{align*}|a_3-\mu a_2^2|\leq|K|+2|M|+|L|=\frac{9\mu}{4}-\frac{5}{3}.\end{align*} $$

The inequality is sharp and the equality holds for the function $f\in \mathcal {K}_{u}$ given by (2.2) and (2.4) with $\omega (z)=z$ and $\rho (z)=z$ , that is,

$$ \begin{align*} f(z)=\frac{2z}{1-z}+\log(1-z)=z+\frac{3}{2}z^2+\frac{5}{3}z^3+\cdots.\\[-43pt] \end{align*} $$

Finally, we establish a result related to the pre-Schwarzian norm for functions in $\mathcal {K}_{u}$ . We first note that a function f in $\mathcal {A}$ belongs to $\mathcal {K}_{u}$ if there exists a function $g\in \mathcal {S}^*$ such that $|zf'(z)/g(z)-1|<1.$ In other words, if there exists a convex function $h\in \mathcal {C}$ with $g(z)=zh'(z)$ such that

$$ \begin{align*}\bigg|\frac{f'(z)}{h'(z)}-1\bigg|<1.\end{align*} $$

Theorem 2.5. Let $f\in \mathcal {K}_{u}$ and $h\in \mathcal {C}$ be the associated convex function. Then,

$$ \begin{align*} |\, \|P_f\|-\|P_h\|\, |\le 2, \end{align*} $$

and the estimate is sharp. Further, $\|P_f\|\le 6$ .

Proof. Let $f\in \mathcal {K}_{u}$ and $h\in \mathcal {C}$ be the associated convex function such that

$$ \begin{align*}\bigg|\frac{f'(z)}{h'(z)}-1\bigg|<1.\end{align*} $$

Then there exists a function $\omega (z)\in \mathcal {B}_{0}$ such that

$$ \begin{align*}\frac{f'(z)}{h'(z)}=1+\omega(z).\end{align*} $$

Taking the logarithmic derivative on both sides,

$$ \begin{align*} \frac{f"(z)}{f'(z)}-\frac{h"(z)}{h'(z)}=\frac{\omega'(z)}{1+\omega(z)} \end{align*} $$

and so,

$$ \begin{align*} \left|\frac{f"(z)}{f'(z)}\right|-\left|\frac{h"(z)}{h'(z)}\right| \leq\left|\frac{f"(z)}{f'(z)}-\frac{h"(z)}{h'(z)}\right| =\left|\frac{\omega'(z)}{1+\omega(z)}\right|. \end{align*} $$

Thus,

$$ \begin{align*} |\, \|P_f\|-\|P_h\| \,|&=\bigg|\sup\limits_{z\in\mathbb{D}}(1-|z|^2)\bigg|\frac{f"(z)}{f'(z)}\bigg| -\sup\limits_{z\in\mathbb{D}}(1-|z|^2)\bigg|\frac{h"(z)}{h'(z)}\bigg|\,\bigg|\\&\le\sup\limits_{z\in\mathbb{D}}(1-|z|^2)\bigg|\bigg(\bigg|\frac{f"(z)}{f'(z)}\bigg|-\bigg|\frac{h"(z)}{h'(z)}\bigg|\bigg)\bigg| \\&\leq\sup\limits_{z\in\mathbb{D}}(1-|z|^2)\bigg|\frac{f"(z)}{f'(z)}-\frac{h"(z)}{h'(z)}\bigg| \\&= \sup\limits_{z\in\mathbb{D}}(1-|z|^2)\frac{|\omega'(z)|}{|1+\omega(z)|}. \end{align*} $$

Since $\omega (z)\in \mathcal {B}_0$ , by the Schwarz–Pick lemma,

$$ \begin{align*} |\omega'(z)|\le\frac{1-|\omega(z)|^2}{1-|z|^2}. \end{align*} $$

Therefore,

$$ \begin{align*} |\, \|P_f\|-\|P_h\| \,|\le \sup\limits_{z\in\mathbb{D}}\frac{1-|\omega(z)|^2}{|1+\omega(z)|}\le 2. \end{align*} $$

The above inequality is sharp for the functions

$$ \begin{align*}f(z)=-\log(1-z)\quad \text{and}\quad h(z)=\frac{z}{1-z}.\end{align*} $$

It is well known that $\|P_h\|\le 4$ for $f\in \mathcal {C}$ (see [19]), and so $\|P_f\|\le 6$ .