Some properties of the class $\mathcal{U}$

TL;DR

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Contribution

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Abstract

In this paper we study the class $\mathcal{U}$ of functions that are analytic in the open unit disk ${\mathbb D}=\{z:|z|<1\}$, normalized such that $f(0)=f'(0)-1=0$ and satisfy \[\left|\left [\frac{z}{f(z)} \right]^{2}f'(z)-1 \right|<1\quad\quad (z\in {\mathbb D}).\] For functions in the class $\mathcal{U}$ we give sharp estimate of the second ant the third Hankel determinant, its relationship with the class of $\alpha$-convex functions, as well as certain starlike properties.

Equations

$$\left|{\left[\frac{z}{f(z)}\right]}^2{f}^{\prime }(z)-1\right|<1(z\in \mathbb{D}).$$

$${\mathcal{S}}^{\star }=\left\{f\in \mathcal{A}:\mathrm{Re}\left[\frac{z{f}^{\prime }(z)}{f(z)}\right]>0,z\in \mathbb{D}\right\}$$

$$\mathcal{K}=\left\{f\in \mathcal{A}:\mathrm{Re}\left[1+\frac{z{f}^{\prime \prime }(z)}{f^{\prime }(z)}\right]>0,z\in \mathbb{D}\right\}.$$

$$z\in f(\mathbb{D})\Rightarrow tz\in f(\mathbb{D})\text{mbox}forallt\in [0,1].$$

$$\mathrm{Re}\left\{(1-\alpha )\frac{z{f}^{\prime }(z)}{f(z)}+\alpha \left[1+\frac{z{f}^{\prime \prime }(z)}{f^{\prime }(z)}\right]\right\}>0,(z\in \mathbb{D}),$$

$$\left|U_f(z)\right|<1(z\in \mathbb{D}),$$

$$U_f(z):={\left[\frac{z}{f(z)}\right]}^2{f}^{\prime }(z)-1.$$

$$H_{q}(n)=\left|\begin{array}[]{cccc}a_{n}&a_{n+1}&\ldots&a_{n+q-1}\\ a_{n+1}&a_{n+2}&\ldots&a_{n+q}\\ \vdots&\vdots&{}\hfil&\vdots\\ a_{n+q-1}&a_{n+q}&\ldots&a_{n+2q-2}\\ \end{array}\right|.$$

$$\begin{array}{rl}{H}_2(2)& =a_2{a}_4-a_3^2,\\ H_3(1)& =a_3(a_2{a}_4-a_3^2)-a_4(a_4-a_2{a}_3)+a_5(a_3-a_2^2).\end{array}$$

$$|H_2(2)|\le 1\text{mbox}and|H_3(1)|\le \frac{1}{4}.$$

$$\frac{z}{f(z)}=1-a_{2}z-z{\int }_0^z\frac{\omega (t)}{t^2}dt,$$

$$|c_1|\le 1,|2c_2|\le 1-|c_1{|}^2\text{mbox}and|3c_3(1-|c_1{|}^2)+4\overline{c_1}{c}_2^2|\le (1-|c_1{|}^2{)}^2-4|c_2{|}^2.$$

$$\begin{array}{rl}f(z)& =\frac{z}{1-\left(a_{2}z+c_1{z}^2+c_2{z}^3+\cdots \right)}\\ & =z+a_2{z}^2+\left(c_1+a_2^2\right)z^3+\left(c_2+2a_2{c}_1+a_2^3\right)z^4\\ & +\left(c_3+2a_2{c}_2+c_1^2+3a_2^2{c}_1+a_2^4\right)z^5\cdots .\end{array}$$

$$a_3=c_1+a_2^2,a_4=c_2+2a_2{c}_1+a_2^3,a_5=c_3+2a_2{c}_2+c_1^2+3a_2^2{c}_1+a_2^4.$$

$$0\le c_1\le 1,|c_2|\le \frac{1}{2}\left(1-c_1^2\right)\text{mbox}and|c_3|\le \frac{1}{3}\left(1-c_1^2-\frac{4|c_2{|}^2}{1+c_1}\right).$$

$$\begin{array}{rl}\left|H_2(2)\right|=& \left|c_2{a}_2-c_1^2\right|\le |c_2|\cdot |a_2|+c_1^2\le \frac{1}{2}\left(1-c_1^2\right)|a_2|+c_1^2\\ =& \frac{1}{2}\cdot |a_2|+\left(1-\frac{1}{2}\cdot |a_2|\right)c_1^2\le 1.\end{array}$$

$$\begin{array}{rl}\left|H_3(1)\right|& =\left|c_1{c}_3-c_2^2\right|\le c_1|c_3|+|c_2{|}^2\\ & \le \frac{1}{3}{c}_1\left(1-c_1^2-\frac{4|c_2{|}^2}{1+c_1}\right)+|c_2{|}^2\\ & =\frac{1}{3}\left(c_1-c_1^3+\frac{3-c_1}{1+c_1}|c_2{|}^2\right)\\ & =\frac{1}{3}\left(c_1-c_1^3+\frac{3-c_1}{1+c_1}\cdot \frac{1}{4}{\left(1-c_1^2\right)}^2\right)\\ & =\frac{1}{12}\left(3-2c_1^2-c_1^4\right)\le \frac{3}{12}=\frac{1}{4}.\end{array}$$

$$2\frac{z{f}^{\prime }(z)}{f(z)}-\left[1+\frac{z{f}^{\prime \prime }(z)}{f^{\prime }(z)}\right]=1-\frac{z{p}^{\prime }(z)}{p(z)+1}.$$

$$\mathrm{Re}\left\{(1+\alpha )\frac{z{f}^{\prime }(z)}{f(z)}-\alpha \left[1-\frac{z{p}^{\prime }(z)}{p(z)+1}\right]\right\}>0,z\in \mathbb{D}.$$

$$\begin{array}{rl}\mathrm{Re}& \left\{(1+\alpha )\frac{z_0{f}^{\prime }(z_0)}{f(z_0)}-\alpha \left[1-\frac{k{e}^{i\theta }}{e^{i\theta }+1}\right]\right\}\\ =& (1+\alpha )\mathrm{Re}\left[\frac{z_0{f}^{\prime }(z_0)}{f(z_0)}\right]+\alpha \frac{k-2}{2}\le 0\end{array}$$

$$g(z)=\frac{z/f(z)-1}{-a_2}$$

$$\left|\frac{z{g}^{\prime }(z)}{g(z)}-1\right|<1(|z|<|a_2|/2);$$

$$\frac{z}{f(z)}=1-a_{2}z-z{\omega }_1(z),$$

$$g(z)=z+\frac{1}{a_2}z{\omega }_1(z).$$

$$\left|\frac{z{g}^{\prime }(z)}{g(z)}-1\right|=\left|\frac{z{\omega }_1^{\prime }(z)}{a_2+{\omega }_1(z)}\right|\le \frac{|z|}{|a_2|-|z|}<1$$

$$\frac{z}{f_b(z)}=1+bz+z^2,0<b\le 2,$$

$$g_b(z)=\frac{\frac{z}{f_b(z)}-1}{b}=z+\frac{1}{b}{z}^2.$$

$$\mathrm{Re}\frac{z{g}_b^{\prime }(z)}{g_b(z)}\ge \frac{1-\frac{2}{b}|z|}{1-\frac{1}{b}|z|}>0.$$

$$|z{\omega }_1^{\prime }(z)-{\omega }_1(z)|\le \frac{r^2-|{\omega }_1(z){|}^2}{1-r^2},$$

$$\begin{array}{rl}|{\mathcal{U}}_g(z)|& =\left|\frac{\frac{1}{a_2}(z{\omega }_1^{\prime }(z)-{\omega }_1(z))-\frac{1}{a_2^2}{\omega }_1^2(z)}{{\left(1+\frac{1}{a_2}{\omega }_1(z)\right)}^2}\right|\\ & \le \frac{|a_2||z{\omega }_1^{\prime }(z)-{\omega }_1(z)|+|{\omega }_1(z){|}^2}{(|a_2|-|{\omega }_1(z)|{)}^2}\\ & \le \frac{|a_2|\frac{r^2-|{\omega }_1(z){|}^2}{1-r^2}+|{\omega }_1(z){|}^2}{(|a_2|-|{\omega }_1(z)|{)}^2}\\ & =:\frac{1}{1-r^2}\phi (t),\end{array}$$

Full text

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For functions in the class $\mathcal{U}$ we give sharp estimate of the second ant the third Hankel determinant, its relationship with the class of $\alpha$-convex functions, as well as certain starlike properties.

1. Introduction

Let ${\mathcal{A}}$ denote the family of all analytic functions in the unit disk ${\mathbb{D}}:=\{z\in{\mathbb{C}}:\,|z|<1\}$ and satisfying the normalization $f(0)=0=f^{\prime}(0)-1$. Let $\mathcal{S}^{\star}$ and $\mathcal{K}$ denote the subclasses of ${\mathcal{A}}$ which are starlike and convex in ${\mathbb{D}}$, respectively, i.e.,

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and

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Geometrical characterisation of convexity is the usual one, while for the starlikeness we have that $f\in\mathcal{S}^{\star}$, if, and only if, $f({\mathbb{D}})$ is a starlike region, i.e.,

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The linear combination of the expressions involved in the analytical representations of starlikeness and convexity brings us to the classes of $\alpha$-convex functions introduced in 1969 by Mocanu ([3]) and consisting of functions $f\in\mathcal{A}$ such that

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where $\frac{f(z)f^{\prime}(z)}{z}\neq 0$ for $z\in{\mathbb{D}}$ and $\alpha\in\mathbb{R}$. Those classes he denoted by $\mathcal{M_{\alpha}}$.

Further, let $\mathcal{U}$ denote the set of all $f\in{\mathcal{A}}$ satisfying the condition

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where the operator $\operatorname{U}_{f}$ is defined by

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All this classes consist of univalent functions and more details on them can be found in [1, 10].

The class of starlike functions is very large and in the theory of univalent functions it is significant if a class doesn’t entirely lie inside $\mathcal{S}^{\star}$. One such case is the class of functions with bounded turning consisting of functions $f$ from $\mathcal{A}$ that satisfy $\operatorname{Re}f^{\prime}(z)>0$ for all $z\in{\mathbb{D}}$. Another example is the class $\mathcal{U}$ defined above and first treated in [5] (see also [6, 7, 10]). Namely, the function $-\ln(1-z)$ is convex, thus starlike, but not in $\mathcal{U}$ because $\operatorname{U}_{f}(0.99)=3.621\ldots>1$, while the function $f$ defined by $\frac{z}{f(z)}=1-\frac{3}{2}z+\frac{1}{2}z^{3}=(1-z)^{2}\left(1+\frac{z}{2}\right)$ is in $\mathcal{U}$ and such that $\frac{zf^{\prime}(z)}{f(z)}=-\frac{2\left(z^{2}+z+1\right)}{z^{2}+z-2}=-\frac{1}{5}+\frac{3i}{5}$ for $z=i$. This rear property is the main reason why the class $\mathcal{U}$ attracts huge attention in the past decades.

In this paper we give sharp estimates of the second and the third Hankel determinant over the class $\mathcal{U}$ and study its relation with the class of $\alpha$-convex and starlike functions.

2. Main results

In the first theorem we give the sharp estimates of the Hankel determinants of second and third order for the class $\mathcal{U}$. We first give the definition of the Hankel determinant, whose elements are the coefficients of a function $f\in{\mathcal{A}}$.

Definition 2**.**

Let $f\in{\mathcal{A}}$. Then the $qth$ Hankel determinant of $f$ is defined for $q\geq 1$, and $n\geq 1$ by

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Thus, the second and the third Hankel determinants are, respectively,

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Theorem 1**.**

Let $f\in\mathcal{U}$ and $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\ldots$. Then we have the sharp estimates:

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Proof.

In [5] the following characterization for functions $f$ in the class in ${\mathcal{U}}$ was given:

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where function $\omega$ is analytic in ${\mathbb{D}}$ with $\omega(0)=\omega^{\prime}(0)=0$ and $|\omega(z)|<1$ for all $z\in{\mathbb{D}}$.

If we put $\omega_{1}(z)=\int_{0}^{z}\frac{\omega(t)}{t^{2}}\,dt$, then we easily obtain that $|\omega_{1}(z)|\leq|z|<1$ and $|\omega_{1}^{\prime}(z)|\leq 1$ for all $z\in{\mathbb{D}}$. If $\omega_{1}(z)=c_{1}z+c_{2}z^{2}+\cdots$, then $\omega_{1}^{\prime}(z)=c_{1}+2c_{2}z+3c_{3}z^{2}+\cdots$ and $|\omega_{1}^{\prime}(z)|\leq 1$, $z\in{\mathbb{D}}$, gives (see relation (13) in the paper of Prokhorov and Szynal [8]):

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Also, from (4) we have

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From the last relation we have

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We may suppose that $c_{1}\geq 0$, since from (6) we have $c_{1}=a_{3}-a_{2}^{2}$ and $a_{3}$ and $a_{2}^{2}$ have the same turn under rotation. In that sense, from (5) we obtain

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If we use (3), (6) and (7), then

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The functions $k(z)=\frac{z}{(1-z)^{2}}$ and $f_{1}(z)=\frac{z}{1-z^{2}}$ show that the estimate is the best possible.

Similarly, after some calculations we also have

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The function $f_{2}(z)=\frac{z}{1-z^{3}/2}$ shows that the result is the best possible. ∎

In the rest of the paper be consider some starlikeness problems for the class $\mathcal{U}$ and its connection with the class of $\alpha$-convex functions.

First, let recall the classical results about the relation between the starlike functions and $\alpha$-convex functions.

Theorem 2**.**

**

• (a)

$\mathcal{M_{\alpha}}\subseteq\mathcal{S}^{\star}$* for every real $\alpha$ $($[4]$)$;*

• (b)

for $0\leq\frac{\beta}{\alpha}\leq 1$ we have $\mathcal{M_{\alpha}}\subset\mathcal{M_{\beta}}$ and for $\alpha>1$, $\mathcal{M_{\alpha}}\subset\mathcal{M}_{1}=\mathcal{K}$ $($[9, 4]$)$.

As an anlogue of the above theorem we have

Theorem 3**.**

For the classes $\mathcal{M_{\alpha}}$ the next results are valid.

• (a)

$\mathcal{M_{\alpha}}\subset\mathcal{U}$* for $\alpha\leq-1$;*

• (b)

$\mathcal{M_{\alpha}}$* is not a subset of $\mathcal{U}$ for any $0\leq\alpha\leq 1$.*

Proof.

• (a)

Let $p(z)=\operatorname{U}_{f}(z).$ Then $p$ is analytic in ${\mathbb{D}}$ and $p(0)=p^{\prime}(0)=0$. From here we have that $\left[\frac{z}{f(z)}\right]^{2}f^{\prime}(z)=p(z)+1$ and, after some calculations that

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The relation (1) is equivalent to

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We want to prove that $|p(z)|<1$, $z\in{\mathbb{D}}$. If not, then according to the Clunie-Jack Lemma ([2]) there exists a $z_{0}$, $|z_{0}|<1$, such that $p(z_{0})=e^{i\theta}$ and $z_{0}p^{\prime}(z_{0})=kp(z_{0})=ke^{i\theta}$, $k\geq 2.$ For such $z_{0}$, from (8) we have that

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since $f\in\mathcal{S}^{\star}$ (by Theorem 2) and $\alpha\leq-1$. That is the contradiction to (1). It means that $|p(z)|=|\operatorname{U}_{f}(z)|<1$, $z\in{\mathbb{D}},$ i.e. $f\in\mathcal{U}.$

• (b)

To prove this part, by using Theorem 2(b), it is enough to find a function $g\in\mathcal{K}$ such that $g$ not belong to the class $\mathcal{U}$. Really, the function $g(z)=-\ln(1-z)$ is convex but not in $\mathcal{U}.$

∎

Open problem. It remains an open problem to study the relationship between classes $\mathcal{M_{\alpha}}$ and $\mathcal{U}$ when $-1<\alpha<0$ and $\alpha>1$.

In the next theorem we consider starlikeness of the function

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where $f\in\mathcal{U}$ and $a_{2}=\frac{f^{\prime\prime}(0)}{2}\neq 0$, i.e., its second coefficient doesn’t vanish.

Namely, we have

Theorem 4**.**

Let $f\in\mathcal{U}$. Then, for the function $g$ defined by (9) we have:

• $(a)$

$|g^{\prime}(z)-1|<1$* for $|z|<|a_{2}|/2$;*

• $(b)$

$g\in\mathcal{S}^{\star}$* in the disc $|z|<|a_{2}|/2$ and even more*

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• $(c)$

$g\in\mathcal{U}$* in the disc $|z|<|a_{2}|/2$ if $0<|a_{2}|\leq 1$.*

The results are best possible.

Proof.

Let $f\in\mathcal{U}$ with $a_{2}\neq 0$ . Then, by using (4), we have that

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where $\omega_{1}$ is analytic in ${\mathbb{D}}$ such that $|\omega_{1}(z)|\leq|z|$ and $|\omega^{\prime}_{1}(z)|\leq 1$. The appropriate function $g$ from (9) has the form

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From here $|g^{\prime}(z)-1|=\frac{1}{|a_{2}|}|\omega_{1}(z)+z\omega_{1}^{\prime}(z)|<1$ for $|z|<|a_{2}|/2$.

By using previous representation, we obtain

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if $|z|<|a_{2}|/2$. It means that the function $g$ is starlike in the disk $|z|<|a_{2}|/2$.

If we consider function $f_{b}$ defined by

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then $f_{b}\in\mathcal{U}$ and

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For this function we easily have that for $|z|<b/2$:

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On the other hand side, since $g^{\prime}_{b}(-b/2)=0$, the function $g_{b}$ is not univalent in a bigger disc, which implies that our result is best possible.

Also, by using (9) and the next estimation for the function $\omega_{1}$:

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(where $|z|=r$ and $|\omega_{1}(z)|\leq r$), after some calculation we get

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where we put

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and $|\omega_{1}(z)|=t$, $0\leq t\leq r$. Since

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because $0<|a_{2}|\leq 1$ and $0\leq t<1$. It means that the function $\varphi$ is an increasing function and that

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Finally we have that

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since $|z|<|a_{2}|/2$. That is implies the second statement of the theorem.

As for sharpness, we can also consider the function $f_{b}$ defined by (10) with $0<b\leq 1$. For $|z|<\frac{b}{2}$ we have

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which implies that $g_{b}$ belongs to the class $\mathcal{U}$ in the disc $|z|<b/2$. ∎

We believe that part (b) of the previous theorem is valid for all $0<|a_{2}|\leq 2$. In that sense we have the next

Conjecture 1**.**

Let $f\in\mathcal{U}$. Then the function $g$ defined by the expression (9) belongs to the class $\mathcal{U}$ in the disc $|z|<|a_{2}|/2$. The result is the best possible.