Certain Extremal Problems on a Classical Family of Univalent Functions

Abstract

Consider the collection $\mathcal{A}$ of analytic functions f defined within the open unit disk $\mathbb{D}$, subject to the conditions $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$. For the parameter $\lambda \in [0,1)$, define the subclass $\mathcal{R}\left(\lambda \right)$ as follows:$\mathcal{R}\left(\lambda \right):=\left\{f\in \mathcal{A}:\mathrm{Re}\left(f^{\prime }\left(z\right)\right)>\lambda ,z\in \mathbb{D}\right\}.$ In this paper, we derive sharp bounds on $\left|z{f}^{\prime }\left(z\right)/f\left(z\right)\right|$ for f in the class $\mathcal{R}\left(\lambda \right)$ and compute the boundary length of $f\left(\mathbb{D}\right)$. Additionally, we investigate the inclusion properties of the sequences of partial sums $f_n\left(z\right)=z+{\sum }_{k=2}^n{a}_k{z}^k$ for functions $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n\in \mathcal{R}\left(\lambda \right)$. Our results extend and refine several classical results in the theory of univalent functions.

1. Introduction

The study of analytic functions and their geometric properties forms the cornerstone of geometric function theory (GFT). A central object of study in GFT is the class of univalent functions, which are analytic functions that map the open unit disk $\mathbb{D}=\{z:|z|<1\}$ onto domains without self-intersections. These functions have profound applications in various areas of mathematics and physics, including fluid dynamics, conformal mappings, and potential theory.

1.1. Univalent Functions

Let $\mathcal{A}$ denote the collection of all analytic functions f defined on $\mathbb{D}$ that satisfy the normalization conditions $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$. A function $f\in \mathcal{A}$ is said to be univalent in $\mathbb{D}$ if it maps distinct points in $\mathbb{D}$ to distinct points in its image, i.e., $f\left(z_1\right)=f\left(z_2\right)$ implies $z_1=z_2$ for $z_1,z_2\in \mathbb{D}$. The class of all such univalent functions is denoted by $\mathcal{S}$. One of the trivial examples of $f\in \mathcal{S}$ is $f\left(z\right)=z$. The other example can be $f\left(z\right)=z/(1-z)$ as

$$\begin{array}{c}\hfill f\left(z_1\right)=f\left(z_2\right)\Rightarrow {\displaystyle \frac{z_1}{1-z_1}}={\displaystyle \frac{z_2}{1-z_2}}\Rightarrow z_1-z_1{z}_2=z_2-z_1{z}_2\Rightarrow z_1=z_2.\end{array}$$

One of the most celebrated examples of a univalent function is the Koebe function, defined as

$$\begin{array}{c}\hfill \kappa \left(z\right)={\displaystyle \frac{z}{{(1-z)}^2}}={\displaystyle \frac{1}{4}}\left({\left({\displaystyle \frac{1+z}{1-z}}\right)}^2-1\right)=z+\sum _{n=2}^{\infty }n{z}^n,z\in \mathbb{D}.\end{array}$$

(1)

The Koebe function maps $\mathbb{D}$ onto the slit domain $\mathbb{C}\setminus (-\infty ,-1/4]$. The function $\kappa \left(z\right)$ and its rotations $e^{-i\theta }k\left(e^{i\theta }z\right)$ play a pivotal role in the theory of univalent functions. In particular, $\kappa \left(z\right)$ serves as an extremal function for many classical problems, including growth, distortion, and rotation. Figure 1 illustrates the rate of growth of the region $\kappa \left(\right|z|<r)$ as r moves from 0 to 1 and shows that $\kappa \left(\right|z|<r)$ blasts out to cover the whole of $\mathbb{C}$ with $(-\infty ,-1/4]$ removed as $r\to 1$.

For the foundational principles and core results in the theory of univalent functions, the works of Duren [1], Pommerenke [2] and Goodman [3] serve as essential references. Additionally, for a comprehensive overview of recent advancements and developments in univalent function theory, the monograph by Thomas et al. [4] provides an up-to-date and thorough exploration of the subject.

For $0\le \lambda <1$, we define two important subclasses of the family of univalent functions in $\mathbb{D}$, namely the families of starlike functions of order λ and convex functions of order λ, denoted by ${\mathcal{S}}^{*}\left(\lambda \right)$ and $\mathcal{C}\left(\lambda \right)$, respectively. These classes are characterized analytically by the conditions

$$f\in {\mathcal{S}}^{*}\left(\lambda \right)\iff \mathrm{Re}\left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right)>\lambda ,z\in \mathbb{D},$$

and

$$\begin{array}{c}\hfill f\in \mathcal{C}\left(\lambda \right)\iff \mathrm{Re}\left(1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)>\lambda ,z\in \mathbb{D}.\end{array}$$

These conditions reflect the geometric properties of the respective function classes: the former ensures that the image of f remains starlike with respect to the origin, while the latter guarantees convexity. Notably, when the order parameter is set to $\lambda =0$, these classes reduce to their classical counterparts, namely the families of starlike functions ${\mathcal{S}}^{*}$ and convex functions $\mathcal{C}$, which have been extensively studied in geometric function theory; see [1,2,3].

Another fundamental subclass of $\mathcal{A}$ is the family of close-to-convex functions of order λ and argument μ, denoted by ${\mathcal{K}}_g(\lambda ,\mu )$. A function $f\in \mathcal{A}$ belongs to ${\mathcal{K}}_g(\lambda ,\mu )$ (see [5,6]) if there exists a function $g\in {\mathcal{S}}^{*}$ and a real parameter $\mu \in (-\pi /2,\pi /2)$, such that

$$\begin{array}{c}\hfill \mathrm{Re}\left(e^{i\mu }{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{g\left(z\right)}}\right)>\lambda ,z\in \mathbb{D}.\end{array}$$

(2)

This definition extends the notion of closeness to convexity by permitting controlled deviations from strict convexity, governed by the parameters $\lambda$ and the reference function g. Notably, when $\lambda =\mu =0$, we retrieve the classical family $\mathcal{K}={\mathcal{K}}_g(0,0)$ of close-to-convex functions, originally introduced by Kaplan [7].

Geometrically, a function f belongs to $\mathcal{K}$ if and only if none of the image curves $C_r:=f\left(\right\{\left|z\right|=r\left\}\right)$ for $0<r<1$ exhibit a reverse hairpin turn, meaning that the image domain remains free from inward-pointing loops that could obstruct its univalence. This criterion provides an intuitive geometric interpretation of closeness to convexity, distinguishing it from the stricter notion of convexity while preserving essential structural properties.

1.2. Functions Whose Derivatives Lie in a Half-Plane

For $0\le \lambda <1$, we introduce $\mathcal{R}\left(\lambda \right)$ as

$$\begin{array}{c}\hfill \mathcal{R}\left(\lambda \right):=\left\{f\in \mathcal{A}:Re\left(f^{\prime }\left(z\right)\right)>\lambda ,z\in \mathbb{D}\right\}.\end{array}$$

(3)

This class, originally introduced by MacGregor [8], consists of analytic functions whose derivatives have real parts bounded below by $\lambda$, ensuring controlled geometric behaviour in the unit disk.

A fundamental property of this family follows from the Noshiro–Warschawski theorem (see [1]), which guarantees that every function in $\mathcal{R}\left(\lambda \right)$ is univalent. Moreover, this class exhibits strong structural properties: it forms a convex and compact subfamily within the class of univalent functions $\mathcal{S}$. A key observation is that when choosing the reference function $g\left(z\right)=z$ and setting $\mu =0$ in (2), the family $\mathcal{R}\left(\lambda \right)$ coincides precisely with the subclass ${\mathcal{K}}_z\left(\lambda \right)$, further reinforcing its geometric significance.

MacGregor [8] conducted a comprehensive analysis of the family $\mathcal{R}:=\mathcal{R}\left(0\right)$, obtaining key results on distortion estimates, the radius of convexity, and the univalence of partial sums. Further contributions were made by Hallenbeck [9], who explored characteristic results for $\mathcal{R}\left(\lambda \right)$, establishing connections between this class and other well-known subclasses of univalent functions. Moreover, some radius problems of this nature have been recently discussed by Todorov [10].

In this paper, we focus on three fundamental problems related to the family $\mathcal{R}\left(\lambda \right)$:

• Sharp estimates for the functional $\left|z{f}^{\prime }\left(z\right)/f\left(z\right)\right|$, where $f\in \mathcal{R}\left(\lambda \right)$ and $z\in \mathbb{D}$.
• Computation of the length of the boundary curve of $f\left(\mathbb{D}\right)$ for $f\in \mathcal{R}\left(\lambda \right)$.
• Inclusion properties of the sequences of partial sums $f_n\left(z\right)=z+{\sum }_{k=2}^n{a}_k{z}^k$ for functions $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n\in \mathcal{R}\left(\lambda \right)$.

Our results extend and refine several classical results in the theory of univalent functions, providing new insights into the geometric and analytic properties of $\mathcal{R}\left(\lambda \right)$.

2. Extremizing the Functional

In this section, we derive sharp estimates for the quantity $\left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right|$ for functions in the family $\mathcal{R}\left(\lambda \right)$. These estimates extend and refine earlier results for the family $\mathcal{R}$, which were partially addressed by [11,12] and later fully resolved by Gray and Ruscheweyh [13]. Our results provide a complete solution for the family $\mathcal{R}\left(\lambda \right)$ and establish sharp bounds for the ratio $\left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right|$.

The problem of estimating $\left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right|$ is of fundamental importance in the theory of univalent functions. For the family $\mathcal{R}$, Gray and Ruscheweyh [13] proved the following sharp result:

Theorem 1

([13]). For $f\in \mathcal{R}$ and $\left|z\right|=r<1$, we have

$${\displaystyle \frac{1-r}{1+r}}\times {\displaystyle \frac{r}{-r+2log(1+r)}}\le \left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right|\le {\displaystyle \frac{1+r}{1-r}}\times {\displaystyle \frac{-r}{r+2log(1-r)}}.$$

The bounds are sharp for the function

$$\kappa \left(z\right)=-z-2log(1-z),$$

with equality achieved at $z=-r$ (lower bound) and $z=r$ (upper bound).

Our goal is to generalize this result to the family $\mathcal{R}\left(\lambda \right)$. To achieve this, we first establish several key lemmas and corollaries that will be instrumental in proving our main theorem.

Lemma 1

([14]). A function $g\left(z\right)$ is analytic in $\mathbb{D}$ with $g\left(0\right)=1$ and $\mathrm{Re}\left(g\left(z\right)\right)>{\displaystyle \frac{1}{2}}$ if and only if

$$g\left(z\right)={\displaystyle \frac{1}{1+z\psi \left(z\right)}},$$

where $\psi \left(z\right)$ is analytic and satisfies $\left|\psi \right(z\left)\right|\le 1$ in $\mathbb{D}$.

Using this lemma, we derive the following estimates for functions with positive real parts.

Theorem 2.

Let $g\left(z\right)$ be analytic in $\mathbb{D}$ and satisfy $g\left(0\right)=1$ and $\mathrm{Re}\left(g\left(z\right)\right)>{\displaystyle \frac{1}{2}}$ for all $z\in \mathbb{D}$. Then,

i .

$\mathrm{Re}\left(g\left(z\right)\right)\le {\displaystyle \frac{1}{1-\left|z\right|}}$ for all $z\in \mathbb{D}$.

ii .

Define

$$\varphi \left(z\right)={\displaystyle \frac{1}{z}}{\int }_0^{z}g\left(\omega \right)d\omega .$$

Then,

$$\mathrm{Re}\left(\varphi \left(z\right)\right)\le {\displaystyle \frac{-1}{\left|z\right|}}log(1-|z\left|\right),z\ne 0.$$

Proof. 

• According to Lemma 1, there exists a function $\psi$ analytic in $\mathbb{D}$ such that $\left|\psi \right(z\left)\right|\le 1$ and

  $$g\left(z\right)={\displaystyle \frac{1}{1+z\psi \left(z\right)}}.$$
  Taking absolute values, we obtain

  $$\left|g\left(z\right)\right|\le {\displaystyle \frac{1}{1-\left|z\psi \right(z\left)\right|}}\le {\displaystyle \frac{1}{1-\left|z\right|}}.$$
  Since $\mathrm{Re}\left(g\right(z\left)\right)\le \left|g\right(z\left)\right|$, the result follows.
• Substituting $\omega =zu$ and using part i., we get

  $$\mathrm{Re}\left(\varphi \left(z\right)\right)=\mathrm{Re}\left({\int }_0^{1}g\left(zu\right)du\right)\le {\int }_0^1{\displaystyle \frac{1}{1-\left|zu\right|}}du.$$
  Evaluating the integral gives

  $${\int }_0^1{\displaystyle \frac{1}{1-\left|zu\right|}}du={\displaystyle \frac{-1}{\left|z\right|}}log(1-|z\left|\right).$$

This completes the proof. □

Corollary 1.

For all $z\in \mathbb{D}$, we have

$$\mathrm{Re}\left({\displaystyle \frac{log(1-z)}{z}}\right)\ge {\displaystyle \frac{log(1-|z\left|\right)}{\left|z\right|}}.$$

Proof. 

Consider the function $g\left(z\right)={\displaystyle \frac{1}{1-z}}$, which satisfies $g\left(0\right)=1$ and $\mathrm{Re}\left(g\left(z\right)\right)>{\displaystyle \frac{1}{2}}$ for all $z\in \mathbb{D}$ (see [1]). Define

$$h\left(z\right)={\displaystyle \frac{1}{z}}{\int }_0^{z}g\left(\omega \right)d\omega ={\displaystyle \frac{-log(1-z)}{z}}.$$

According to Theorem 2(ii), we have

$$\mathrm{Re}\left(h\left(z\right)\right)=\mathrm{Re}\left({\displaystyle \frac{-log(1-z)}{z}}\right)\le {\displaystyle \frac{-1}{\left|z\right|}}log(1-|z\left|\right).$$

Taking the negative of both sides provides the desired result. □

Definition 1 

(Subordination [1]). ” Let f and g be analytic functions in the open unit disk $\mathbb{D}$. We say that f is subordinate to g in $\mathbb{D}$, denoted by $f\prec g$, if there exists an analytic function $w:\mathbb{D}\to \mathbb{D}$ such that $w\left(0\right)=0$ and $\left|w\right(z\left)\right|\le 1$ for all $z\in \mathbb{D}$, satisfying

$$f\left(z\right)=g\left(w\left(z\right)\right),z\in \mathbb{D}.$$

In particular, if g is univalent in $\mathbb{D}$, then subordination is equivalent to

$$f\prec g\iff f\left(0\right)=g\left(0\right)\mathit{and}f\left(\mathbb{D}\right)\subset g\left(\mathbb{D}\right).$$

This characterization highlights that under univalence, subordination implies that f preserves both the initial value and the image inclusion structure dictated by g.

Lemma 2. 

Let $f\in \mathcal{R}\left(\lambda \right)$. For all $\left|z\right|<1$, we have

$$\begin{array}{c}\hfill \mathrm{Re}\left({\displaystyle \frac{f\left(z\right)}{z}}\right)\ge (2\lambda -1)+2(\lambda -1){\displaystyle \frac{log(1-|z\left|\right)}{\left|z\right|}}.\end{array}$$

(4)

Proof. 

Since $f\in \mathcal{R}\left(\lambda \right)$, it follows from [9] (Theorem 6) that

$$\begin{array}{c}\hfill {\displaystyle \frac{f\left(z\right)}{z}}\prec g\left(z\right):=(2\lambda -1)+2(\lambda -1){\displaystyle \frac{log(1-z)}{z}},z\in \mathbb{D}.\end{array}$$

(5)

The function $g\left(z\right)$ is univalent in $\mathbb{D}$ since ${\displaystyle \frac{log(1-z)}{z}}$ is univalent in $\mathbb{D}$. According to the subordination principle, (see [1] (Chapter 6)), we obtain

$$\mathrm{Re}\left({\displaystyle \frac{f\left(z\right)}{z}}\right)\ge (2\lambda -1)+2(\lambda -1)\mathrm{Re}\left({\displaystyle \frac{log(1-z)}{z}}\right).$$

Applying Corollary 1 to the right-hand side completes the proof. □

We now present our main theorem, which establishes sharp bounds for $\left|z{f}^{\prime }\left(z\right)/f\left(z\right)\right|,$ where $f\in \mathcal{R}\left(\lambda \right)$.

Theorem 3.

For $f\in \mathcal{R}\left(\lambda \right)$ and $\left|z\right|=r<1$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1-(1-2\lambda )r}{1+r}}{\mathcal{L}}_{\mathcal{R}}(\lambda ,r)\le \left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right|\le {\displaystyle \frac{1+(1-2\lambda )r}{1-r}}{\mathcal{U}}_{\mathcal{R}}(\lambda ,r),\end{array}$$

(6)

where

$${\mathcal{L}}_{\mathcal{R}}(\lambda ,r)={\displaystyle \frac{r}{-(1-2\lambda )r+2(1-\lambda )log(1+r)}},$$

and

$${\mathcal{U}}_{\mathcal{R}}(\lambda ,r)={\displaystyle \frac{-r}{(1-2\lambda )r+2(1-\lambda )log(1-r)}}.$$

The bounds are sharp for the function

$${\kappa }_{\lambda }\left(z\right)=(2\lambda -1)z+2(\lambda -1)log(1-z),$$

with equality achieved at $z=-r$ (lower bound) and $z=r$ (upper bound).

Proof. 

We note that the family $\mathcal{R}\left(\lambda \right)$ is compact and rotationally invariant (see [9]), i.e., if $f\in \mathcal{R}\left(\lambda \right)$, then $f\left(xz\right)/x\in \mathcal{R}\left(\lambda \right)$ for all x with $\left|x\right|=1$. Therefore, in view of [15] (Theorem 1.1, Corollary 1.1, Theorem 1.6), it is sufficient to prove the inequality (6) for the extremal function $f\left(z\right)={\kappa }_{\lambda }\left(z\right)=(2\lambda -1)z+2(\lambda -1)log(1-z).$ We will first establish the lower bound by making use of the following mixed form of the triangle inequalities:

$$\begin{array}{c}\hfill {\displaystyle \frac{1-\left|az\right|}{1+\left|bz\right|}}\le \left|{\displaystyle \frac{1\pm az}{1\pm bz}}\right|\le {\displaystyle \frac{1+\left|az\right|}{1-\left|bz\right|}},\mathrm{for}\left|az\right|<1\mathrm{and}\left|bz\right|<1.\end{array}$$

(7)

Since ${\kappa }_{\lambda }\left(z\right)$ is analytic and ${\kappa }_{\lambda }\left(0\right)=0$, we can express $f\left(z\right)={\kappa }_{\lambda }\left(z\right)$ using the Fundamental Theorem of Calculus as

$$f\left(z\right)={\int }_0^1{\displaystyle \frac{d}{ds}}\left(f\left(sz\right)\right)ds.$$

Applying the chain rule, this simplifies to

$$f\left(z\right)=z{\int }_0^1{f}^{\prime }\left(sz\right)ds.$$

Since $f\left(0\right)=1={lim}_{z\to 0}{\displaystyle \frac{f\left(z\right)}{z}}$, we obtain

$${\displaystyle \frac{f\left(z\right)}{z{f}^{\prime }\left(z\right)}}={\int }_0^1{\displaystyle \frac{f^{\prime }\left(sz\right)}{f^{\prime }\left(z\right)}}ds.$$

Substituting $f^{\prime }\left(sz\right)$ and $f^{\prime }\left(z\right)$, we arrive at

$$\begin{array}{c}\hfill {\displaystyle \frac{f\left(z\right)}{z{f}^{\prime }\left(z\right)}}={\int }_0^1{\displaystyle \frac{1+(1-2\lambda )sz}{1-sz}}·{\displaystyle \frac{1-z}{1+(1-2\lambda )z}}ds.\end{array}$$

(8)

For $0<\lambda <1$ and $z\in \mathbb{D}$, we have $\left|\right(1-2\lambda \left)z\right|<1$. Taking the modulus on (8) and applying the right-hand side of inequality (7), we obtain

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{f\left(z\right)}{z{f}^{\prime }\left(z\right)}}\right|& =\left|{\int }_0^1{\displaystyle \frac{1+(1-2\lambda )sz}{1-sz}}·{\displaystyle \frac{1-z}{1+(1-2\lambda )z}}ds\right|\hfill \\ \hfill & \le {\int }_0^1\left|{\displaystyle \frac{1+(1-2\lambda )sz}{1-sz}}\right|\left|{\displaystyle \frac{1-z}{1+(1-2\lambda )z}}\right|ds\hfill \\ \hfill & \le {\int }_0^1{\displaystyle \frac{1+\left|\right(1-2\lambda \left)z\right|s}{1-s\left|z\right|}}·{\displaystyle \frac{1+\left|z\right|}{1-\left|\right(1-2\lambda \left)z\right|}}ds\hfill \\ \hfill & ={\displaystyle \frac{1+\left|z\right|}{1-\left|\right(1-2\lambda \left)z\right|}}{\int }_0^1{\displaystyle \frac{1+\left|\right(1-2\lambda \left)z\right|s}{1-s\left|z\right|}}ds.\hfill \end{array}$$

Over $\left|z\right|=r<1$, we apply the Maximum Modulus Principle to obtain

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{f\left(z\right)}{z{f}^{\prime }\left(z\right)}}\right|\le {\displaystyle \frac{1+r}{1-(1-2\lambda )r}}{\int }_0^1{\displaystyle \frac{1+(1-2\lambda )rs}{1-sr}}ds,\end{array}$$

(9)

where z is chosen such that $1+\left|\right(1-2\lambda \left)z\right|$ is maximized and $1-\left|\right(1-2\lambda \left)z\right|$ is minimized. The integral in (9) is evaluated as $1/{\mathcal{L}}_{\mathcal{R}}(\lambda ,r)$, leading to

$$\begin{array}{c}\hfill \left|{\displaystyle \frac{f\left(z\right)}{z{f}^{\prime }\left(z\right)}}\right|\le {\displaystyle \frac{1+r}{1-(1-2\lambda )r}}·{\displaystyle \frac{1}{{\mathcal{L}}_{\mathcal{R}}(\lambda ,r)}}.\end{array}$$

(10)

Taking the reciprocal of (10), we obtain the desired lower bound of (6).Now, for the upper bound of (6), using $f\left(z\right)={\kappa }_{\lambda }\left(z\right)=(2\lambda -1)z+2(\lambda -1)log(1-z)$, for $\left|z\right|=r$, we get

$$\begin{array}{c}\hfill \left|f^{\prime }\left(z\right)\right|=\left|{\displaystyle \frac{1+(1-2\lambda )z}{1-z}}\right|\le {\displaystyle \frac{1+(1-2\lambda )\left|z\right|}{1-\left|z\right|}}={\displaystyle \frac{1+(1-2\lambda )r}{1-r}}.\end{array}$$

(11)

Using (11) and (4), we obtain

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right|& =\left|{\displaystyle \frac{f^{\prime }\left(z\right)}{f\left(z\right)/z}}\right|\hfill \\ \hfill & \le {\displaystyle \frac{|f^{\prime }\left(z\right)|}{\mathrm{Re}\left(\frac{f\left(z\right)}{z}\right)}}\hfill \\ \hfill & \le {\displaystyle \frac{1+(1-2\lambda )r}{1-r}}\times {\displaystyle \frac{1}{(2\lambda -1)+2(\lambda -1)\left(\frac{log(1-r)}{r}\right)}}\hfill \\ \hfill & ={\displaystyle \frac{1+(1-2\lambda )r}{1-r}}\times {\displaystyle \frac{-r}{(1-2\lambda )r+2(1-\lambda )log(1-r)}}.\hfill \end{array}$$

This establishes the upper bound in (6).

The bounds are sharp, since the extremal function ${\kappa }_{\lambda }\left(z\right)$ achieves equality at $z=-r$ (lower bound) and $z=r$ (upper bound).

Thus, the proof of the theorem is complete. □

Remark 1.

If we set $\lambda =0$ in Theorem 3, the class $\mathcal{R}\left(\lambda \right)$ reduces to $\mathcal{R}$ (where $\lambda =0$), and the bounds in Theorem 3 reduce to the bounds in Theorem 1.

Corollary 2.

For $f\in \mathcal{R}\left(\lambda \right)$ and $\left|z\right|=r<1$, we have

$$\left|{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right|\le {\displaystyle \frac{-r+(1-2\lambda )r(1-2r)+2(1+\lambda )log(1-r)}{(1-r)\left[\right(1-2\lambda )r+2(1-\lambda )log(1-r\left)\right]}}.$$

The bound is sharp for the extremal function ${\kappa }_{\lambda }\left(z\right)=(2\lambda -1)z+2(\lambda -1)log(1-z)$.

Proof. 

To prove this corollary, we apply Theorem 3 to the function $g\left(z\right)=z{f}^{\prime }\left(z\right)$. Since $f\in \mathcal{R}\left(\lambda \right)$, it follows that $g\left(z\right)$ satisfies

$$\mathrm{Re}\left(g^{\prime }\left(z\right)\right)=\mathrm{Re}(f^{\prime }\left(z\right)+z{f}^{″}\left(z\right))>\lambda .$$

Thus, $g\left(z\right)$ belongs to a class of functions similar to $\mathcal{R}\left(\lambda \right)$, and we can apply the bounds from Theorem 3 to $g\left(z\right)$.

From Theorem 3, we have

$$\left|{\displaystyle \frac{z{g}^{\prime }\left(z\right)}{g\left(z\right)}}\right|\le {\displaystyle \frac{1+(1-2\lambda )r}{1-r}}·{\displaystyle \frac{-r}{(1-2\lambda )r+2(1-\lambda )log(1-r)}}.$$

Substituting $g\left(z\right)=z{f}^{\prime }\left(z\right)$ and $g^{\prime }\left(z\right)=f^{\prime }\left(z\right)+z{f}^{″}\left(z\right)$, we obtain the following result:

$$\left|{\displaystyle \frac{z(f^{\prime }\left(z\right)+z{f}^{″}\left(z\right))}{z{f}^{\prime }\left(z\right)}}\right|\le {\displaystyle \frac{1+(1-2\lambda )r}{1-r}}·{\displaystyle \frac{-r}{(1-2\lambda )r+2(1-\lambda )log(1-r)}}.$$

Simplifying the left-hand side, we get

$$\left|1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right|\le {\displaystyle \frac{1+(1-2\lambda )r}{1-r}}·{\displaystyle \frac{-r}{(1-2\lambda )r+2(1-\lambda )log(1-r)}}.$$

Rearranging terms, we isolate $\left|{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right|$

$$\left|{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right|\le {\displaystyle \frac{1+(1-2\lambda )r}{1-r}}·{\displaystyle \frac{-r}{(1-2\lambda )r+2(1-\lambda )log(1-r)}}-1.$$

Simplifying further, we obtain the desired bound

$$\left|{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right|\le {\displaystyle \frac{-r+(1-2\lambda )r(1-2r)+2(1+\lambda )log(1-r)}{(1-r)\left[\right(1-2\lambda )r+2(1-\lambda )log(1-r\left)\right]}}.$$

The bound is sharp for the extremal function ${\kappa }_{\lambda }\left(z\right)=(2\lambda -1)z+2(\lambda -1)log(1-z)$, as it achieves equality in the distortion estimates used in the proof. □

Remark 2.

This corollary provides a distortion estimate for the second derivative of functions in $\mathcal{R}\left(\lambda \right)$. It generalizes similar results for the class $\mathcal{R}$ (where $\lambda =0$) and demonstrates how the parameter λ influences the growth of $f^{″}\left(z\right)$. The sharpness of the bound highlights the role of the extremal function ${\kappa }_{\lambda }\left(z\right)$ in solving extremal problems for this class.

3. Arc Length of Image Curve

In this section, we investigate the arc length of the image curve $f\left(\right|z|=r)$ for functions $f\in \mathcal{R}\left(\lambda \right)$. The arc length problem is a classical topic in geometric function theory, with significant contributions from MacGregor [8], who established sharp bounds for the family $\mathcal{R}$. Our goal is to extend these results to the family $\mathcal{R}\left(\lambda \right)$ and derive sharp estimates for the arc length of the boundary curve $\partial \left(f\right(\mathbb{D}\left)\right)$.

The arc length $L_r\left(f\right)$ of the image curve $f\left(\right|z|=r)$ is defined as the length of the boundary of the region $f\left(\mathbb{D}\right)$. For any member f of $\mathcal{R}$, MacGregor [8] established that

$$L_r\left(f\right)\le L_r\left(f_0\right),$$

where $f_0\left(z\right)=-z-2log(1-z)$. This result is sharp, and the extremal function $f_0$ plays a crucial role in understanding the geometric properties of $\mathcal{R}$.

In this section, we generalize MacGregor’s result to the family $\mathcal{R}\left(\lambda \right)$. Specifically, we derive sharp upper bounds for $L_r\left(f\right)$ when $f\in \mathcal{R}\left(\lambda \right)$, and we establish the sharpness of these bounds.

To prove our main result, we rely on the following lemma, which provides a key inequality for subordinated functions.

Lemma 3

([1]). Let F and G be analytic in the unit disk $\mathbb{D}$, and suppose $F\prec G$. Then, for $0<p<\infty$,

$${\int }_0^{2\pi }|F\left(r{e}^{i\theta }\right){|}^{p}d\theta \le {\int }_0^{2\pi }{\left|G\left(r{e}^{i\theta }\right)\right|}^{p}d\theta ,0\le r<1.$$

This lemma allows us to compare the integrals of subordinated functions, which is essential for deriving bounds on the arc length.

We now state and prove our main theorem, which provides sharp upper bounds for the arc length $L_r\left(f\right)$ when $f\in \mathcal{R}\left(\lambda \right)$.

Theorem 4.

Let $f\in \mathcal{R}\left(\lambda \right)$, and let $L_r\left(f\right)$ denote the length of the image curve $f\left(\right|z|=r)$ for $0<r<1$. Then,

$$L_r\left(f\right)\le 2\pi r\left[1+{\displaystyle \frac{2(1-\lambda )}{\pi }}log\left({\displaystyle \frac{1+r}{1-r}}\right)\right].$$

The result is sharp.

Proof. 

Let $f\in \mathcal{R}\left(\lambda \right)$. By definition, $\mathrm{Re}\left(f^{\prime }\left(z\right)\right)>\lambda$, which implies the subordination

$$f^{\prime }\left(z\right)\prec {\displaystyle \frac{1+(1-2\lambda )z}{1-z}}.$$

Setting $z=r{e}^{i\theta }$ and applying Lemma 3 with $p=1$, we obtain

$${\int }_0^{2\pi }\left|f^{\prime }\left(r{e}^{i\theta }\right)\right|d\theta \le {\int }_0^{2\pi }\left|{\displaystyle \frac{1+(1-2\lambda )r{e}^{i\theta }}{1-r{e}^{i\theta }}}\right|d\theta .$$

Since $L_r\left(f\right)={\int }_0^{2\pi }\left|f^{\prime }\left(r{e}^{i\theta }\right)\right|rd\theta$, it follows that

$$L_r\left(f\right)\le r{\int }_0^{2\pi }\left|{\displaystyle \frac{1+(1-2\lambda )r{e}^{i\theta }}{1-r{e}^{i\theta }}}\right|d\theta .$$

Expressing the integrand in terms of its real and imaginary parts, we have

$$\left|{\displaystyle \frac{1+(1-2\lambda )r{e}^{i\theta }}{1-r{e}^{i\theta }}}\right|=\left|\lambda +{\displaystyle \frac{(1-\lambda )(1-r^2)}{1-2rcos\theta +r^2}}+i{\displaystyle \frac{2(1-\lambda )rsin\theta }{1-2rcos\theta +r^2}}\right|.$$

Using the properties of the Poisson kernel (see [16] (Chapter 11)) and the fact that

$${\int }_0^{2\pi }{\displaystyle \frac{1-r^2}{1-2rcos\theta +r^2}}d\theta =2\pi ,$$

and

$${\int }_0^{2\pi }{\displaystyle \frac{r|sin\theta |}{1-2rcos\theta +r^2}}d\theta =2log\left({\displaystyle \frac{1+r}{1-r}}\right),$$

we conclude that

$$L_r\left(f\right)\le 2\pi r\left[1+{\displaystyle \frac{2(1-\lambda )}{\pi }}log\left({\displaystyle \frac{1+r}{1-r}}\right)\right].$$

The sharpness of the bound follows from the extremal functions

$$f^{\prime }\left(z\right)={\displaystyle \frac{1+(1-2\lambda )xz}{1-xz}},\left|x\right|=1.$$

This establishes the result completely. □

Remark 3.

When $\lambda =0$, the family $\mathcal{R}\left(\lambda \right)$ reduces to $\mathcal{R}$, and we recover the following classical result as a corollary.

Corollary 3.

For $f\in \mathcal{R}$, the arc-length $L_r\left(f\right)$ satisfies

$$L_r\left(f\right)\le 2\pi r+4rlog\left({\displaystyle \frac{1+r}{1-r}}\right).$$

This result aligns with MacGregor’s original theorem and demonstrates the consistency of our generalized bounds with prior work.

4. Sequence of Partial Sums

In this section, we investigate the properties of the partial sums (or sections) of functions in the family $\mathcal{R}\left(\lambda \right)$. Specifically, we determine the radius of univalence for the nth partial sum $f_n$ of a function $f\in \mathcal{R}\left(\lambda \right)$ and establish conditions under which $f_n$ preserves the properties of the original function f.

The study of partial sums of univalent functions has a rich history in geometric function theory. A fundamental result due to Szegő [17] states that every partial sum of a function $f\in \mathcal{S}$ is univalent in the disk $\left|z\right|<1/4$, and this radius cannot be improved. This is demonstrated by the second partial sum of the Koebe function $\kappa \left(z\right)={\displaystyle \frac{z}{{(1-z)}^2}}$, which fails to be univalent outside this disk.

For functions in $\mathcal{R}\left(\lambda \right)$, the situation is more nuanced. While the partial sums of $f\in \mathcal{R}\left(\lambda \right)$ may not preserve univalence in the entire unit disk $\mathbb{D}$, we can determine a radius $r_n$ such that $f_n$ remains in $\mathcal{R}\left(\lambda \right)$ for $\left|z\right|<r_n$. This section establishes such a result and provides explicit bounds for $r_n$.

We begin by defining the nth partial sum of a function $f\in \mathcal{A}$.

Definition 2.

Let $f\in \mathcal{A}$ be given by

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

The nth partial sum (or nth section) of f, denoted by $f_n$, is the polynomial defined as

$$f_n\left(z\right)=z+\sum _{k=2}^n{a}_k{z}^k.$$

For a given analytic function $f\left(z\right)$ with its partial sum $f_n\left(z\right)$, the partial sum radius $R_n$ is the largest radius, such that $f_n\left(z\right)$ retains a desired geometric property (e.g., univalence, starlikeness or convexity) within the disk $\left|z\right|<R_n$. Beyond this radius, the truncated series may fail to maintain these properties, emphasizing the significance of $R_n$ in geometric function theory. Our main result establishes the radius of univalence for the partial sums $f_n$ of functions in $\mathcal{R}\left(\lambda \right)$.

Theorem 5.

Let $f\in \mathcal{R}\left(\lambda \right)$, and let $f_n$ be its nth partial sum. Then, $f_n\in \mathcal{R}\left(\lambda \right)$ inside the disk $\left|z\right|=R_n$, where

$$R_n=1-{\displaystyle \frac{2logn}{n}},\mathrm{for}n\ge 3.$$

Proof. 

Since $f\in \mathcal{R}\left(\lambda \right)$, we have $\mathrm{Re}\left(f^{\prime }\left(z\right)\right)>\lambda$. Define the function $g\left(z\right)$ as

$$\begin{array}{c}\hfill g\left(z\right)={\displaystyle \frac{f^{\prime }\left(z\right)-\lambda }{1-\lambda }}.\end{array}$$

(12)

It is straightforward to verify that $g\left(z\right)$ is analytic in $\mathbb{D}$, satisfies $\mathrm{Re}\left(g\right(z\left)\right)>0$ for all $z\in \mathbb{D}$, and attains the value $g\left(0\right)=1$. Consequently, $g\left(z\right)$ is a function with a positive real part (such functions are called Carathéodory functions) and, according to a classical result (see [3] (Chapter 7, pp. 77–81)), it admits the series representation

$$g\left(z\right)=1+\sum _{n=1}^{\infty }{b}_n{z}^n,$$

where the coefficients satisfy

$$|b_n|\le 2\mathrm{for}\mathrm{all}n\ge 1.$$

Furthermore, the function $g\left(z\right)$ satisfies the following bound on its real part (see Herglotz’s representation theorem [1] (p. 22) and [3] (Chapter 7, Theorem 4)),

$${\displaystyle \frac{1-r}{1+r}}\le \mathrm{Re}\left(g\left(z\right)\right)\le {\displaystyle \frac{1+r}{1-r}},\mathrm{for}\left|z\right|=r<1.$$

Let $g_n\left(z\right)$ denote the nth partial sum of $g\left(z\right)$, given by

$$g_n\left(z\right)={\displaystyle \frac{f_n^{\prime }\left(z\right)-\lambda }{1-\lambda }}.$$

Now, consider the difference $g_n\left(z\right)=g\left(z\right)-{\sigma }_n\left(z\right)$, where

$${\sigma }_n\left(z\right)=\sum _{k=n+1}^{\infty }{b}_k{z}^k.$$

satisfies

$$|{\sigma }_n\left(z\right)|\le \sum _{k=n+1}^{\infty }|b_k{\left|\right|z|}^k\le 2\sum _{k=n+1}^{\infty }{r}^k=2{\displaystyle \frac{r^{n+1}}{1-r}}\mathrm{for}\left|z\right|=r<1.$$

Therefore,

$$\mathrm{Re}\left(g_n\left(z\right)\right)\ge \mathrm{Re}\left(g\left(z\right)\right)-\left|{\sigma }_n\left(z\right)\right|\ge {\displaystyle \frac{1-r}{1+r}}-{\displaystyle \frac{2r^{n+1}}{1-r}}.$$

To simplify this, we have

$$\mathrm{Re}\left(g_n\left(z\right)\right)\ge {\displaystyle \frac{{(1-r)}^2-2r^{n+1}(1+r)}{1-r^2}}.$$

This expression is non-negative, as shown below:

$${(1-r)}^2-2r^{n+1}(1+r)\ge 0.$$

Let $1-r={\displaystyle \frac{t}{n}}$. Then, $\mathrm{Re}\left(g_n\left(z\right)\right)\ge 0$ if

$$F\left(t\right)={\displaystyle \frac{t^2}{n^2}}-2{\left(1-{\displaystyle \frac{t}{n}}\right)}^{n+1}\left(2-{\displaystyle \frac{t}{n}}\right)\ge 0.$$

Expanding $F\left(t\right)$, we find

$$F\left(t\right)>{\displaystyle \frac{t^2}{n^2}}-2\left(1-{\displaystyle \frac{t}{n}}\right)\left(2-{\displaystyle \frac{t}{n}}\right)e^{-t}.$$

For $t=2logn$, the right-hand side of the above inequality is positive for $n\ge 3$. Thus, $\mathrm{Re}\left(g_n\left(z\right)\right)>0$ for $r=1-(2logn)/n$. Since $\mathrm{Re}\left(g_n\left(z\right)\right)$ is harmonic, we conclude via the maximum modulus principle that $\mathrm{Re}\left(g_n\left(z\right)\right)>0$ in the disk $\left|z\right|<1-(2logn)/n$. Consequently, it follows from (12) that $\mathrm{Re}\left(f_n^{\prime }\left(z\right)\right)>\lambda$ in the disk $\left|z\right|<1-(2logn)/n$. Thus, $f_n\in \mathcal{R}\left(\lambda \right)$ in $\left|z\right|<R_n$ where $R_n=1-(2logn)/n$. □

Remark 4.

The result demonstrates that the partial sums $f_n$ of functions in $\mathcal{R}\left(\lambda \right)$ retain the property $\mathrm{Re}\left(f_n^{\prime }\left(z\right)\right)>\lambda$ in a disk of radius $1-{\displaystyle \frac{2logn}{n}}$. This radius decreases as n increases, reflecting the fact that higher-order partial sums are less likely to preserve the properties of the original function. The result aligns with Szegő’s [17] classical theorem for $\mathcal{S}$ and extends it to the family $\mathcal{R}\left(\lambda \right)$.

5. Conclusions

In this paper, we have investigated several extremal problems for the family $\mathcal{R}\left(\lambda \right)$, consisting of analytic functions with derivatives which have positive real parts bounded by $\lambda$, as shown below. For $\mathcal{R}\left(\lambda \right)$, we determine the estimates for the quantity $\left|z{f}^{\prime }\left(z\right)/f\left(z\right)\right|$, compute the length of the boundary curve of the image domain $f\left(\mathbb{D}\right)$, and find the radius of univalence for the partial sums $f_n$. These results not only generalize and refine existing theorems but also provide new insights into the geometric and analytic properties of $\mathcal{R}\left(\lambda \right)$. Future work could explore similar problems for other subclasses of univalent functions or investigate the behaviour of higher-order partial sums and their geometric implications.