1 Introduction and main result
We denote the set of all complex numbers by ${\mathbb C}$ . Let $\mathcal {A}$ be the class of analytic functions in the unit disc ${\mathbb D}:=\{z\in {\mathbb C}: |z|<1\}$ with the Taylor expansion
and $f(z)\neq 0$ for $z\in {\mathbb D}\setminus \{0\}$ . We see that the functions in $\mathcal {A}$ must satisfy the normalisation $f(0)=0=f'(0)-1$ . For $f\in \mathcal {A}$ , we define
and for $f\in \mathcal {A}$ that are bounded in ${\mathbb D}$ , let
Lewin obtained the following result.
Theorem 1.1 (Lewin, [Reference Lewin4])
For $f\in \mathcal {A}$ with the expansion (1.1), $d_f\leq \exp (-|a_2|/2)$ . If f is bounded, then $D_f\geq \exp (|a_2|/2)$ .
In the same article, Lewin established that the estimates in Theorem 1.1 are best possible. He commented that although the bounds in Theorem 1.1 are not sharp in the case of univalent or bounded univalent functions, they nevertheless supply information which may be of help when dealing with conformal mappings (analytic and univalent mappings).
In this article, we allow the functions in $\mathcal {A}$ to possess a nonzero simple pole inside ${\mathbb D}$ and wish to see whether an analogue of Theorem 1.1 can be established after suitably defining the quantities $d_f$ and $D_f$ in this case. Therefore, we consider functions f that are meromorphic having a simple pole at $z=p\in (0,1)$ inside the unit disk ${\mathbb D}$ , with the Taylor series expansion
where ${\mathbb D}_p:=\{z\in {\mathbb C} : |z|<p\}$ and such that f does not vanish in ${\mathbb D}$ other than at the origin. Evidently, for such f, we have $f(0)=0=f'(0)-1$ . We denote the class of such functions by $\mathcal {F}(p)$ . If g is a meromorphic function having a simple pole at $pe^{i\beta }$ , $\beta \in (0, 2\pi ]$ , $p\in (0,1)$ , and g is nonvanishing in ${\mathbb D}\setminus \{0\}$ with $g(0)=0$ and $g'(0)\neq 0$ , then
This shows that taking the pole p in the interval $(0,1)$ is sufficiently general. For $f\in \mathcal {F}(p)$ , we define
and if $(z-p)f$ is bounded in ${\mathbb D}$ , we define
These quantities can be thought of as analogous to $d_f$ and $D_f$ in [Reference Lewin4]. The reason for multiplying $f/z$ , $f\in \mathcal {F}(p)$ , by the factor ${(z-p)}{(1-pz)}$ is to make the resulting function holomorphic in ${\mathbb D}$ . In addition, if f has a holomorphic extension to the boundary $\partial {\mathbb D}=\{z\in {\mathbb C}: |z|=1\}$ of ${\mathbb D}$ , then
Thus, in such cases, finding the bounds of $d_p(f)$ and $D_p(f)$ will essentially give estimates for the distance between the origin and the image of the unit circle under f. In the second part of this paper, we will generalise these results to functions having more than one simple pole in ${\mathbb D}$ .
We now state and prove our main result. We will adopt the main idea of the proof from [Reference Lewin4], but as we approach the problem, we will realise that the proof itself and finding the extremal functions for which equalities hold in these estimates are not straightforward.
Theorem 1.2. Let $f\in \mathcal {F}(p)$ have the expansion (1.2) in ${\mathbb D}_p$ . Then
and if $(z-p)f$ is bounded in ${\mathbb D}$ , then
These bounds are best possible.
Proof. Let $s>1$ be such that
Then we must have
where we choose that branch of logarithm for which $\log f'(0)=0$ . A minor simplification of the above inequality yields
Now define
which is analytic in ${\mathbb D}$ by choosing that branch of the logarithm for which $\log (f'(0))=0$ . By virtue of the previous inequality, we have $\mathrm {Re}\, F(z)\geq 0$ with $F(0)=1$ . Now we can expand F about the origin to get
An application of Caratheodory’s lemma (see [Reference Carathéodory3]) for the function F in ${\mathbb D}_p$ yields
Letting $d_p(f)=p/s$ gives the first estimate of the theorem. To obtain the second estimate of the theorem, we let
Note that $g\in \mathcal {F}(p)$ as $(z-p)f$ is bounded in ${\mathbb D}$ and g has the Taylor expansion
We thus have $d_p(g)/p=p/D_p(f)$ . Therefore,
Consequently, the second inequality of the theorem follows.
The bounds obtained in the theorem are best possible in the following sense. We consider the functions $f^{\pm }_{\alpha }$ in $\mathcal {F}(p)$ :
where $\psi (z)=z(1-pz)/(p-z)$ . A quick computation yields
Therefore, here $a_2=\alpha /p+1/p-p$ , which gives $|a_2p+p^2-1|=\alpha $ and
where $\phi \in (0, \pi )\cup (\pi , 2\pi )$ . Again for the function $f^{-}_{\alpha }$ , we have
which gives $a_2=-\alpha /p+1/p-p$ or equivalently $|a_2p+p^2-1|=\alpha $ and
where $\phi \in (0, \pi )\cup (\pi , 2\pi )$ . This shows that the estimates stated in the theorem are best possible. This completes the proof of the theorem.
Remark 1.3. (i) Note that the quantity $|pa_2+p^2-1|$ in the bounds for $d_p(f)$ and $D_p(f)$ in Theorem 1.2, may be replaced by $|p^2(a_3-\tfrac 12a_2^2)+(p^4-1)/2|$ as by Caratheodory’s lemma, we also have
for the function F defined in ${\mathbb D}_p$ . Furthermore, we comment here that if $pa_2+p^2- 1=0$ , then we need to use the first nonvanishing coefficient in the expansion (1.4) to get the estimates for $d_p(f)$ and $D_p(f)$ .
(ii) We observe that we recover Lewin’s results (compare [Reference Lewin4, Theorem A]) if we pass to the limit as $p\rightarrow 1-$ in the expression for the bounds obtained in Theorem 1.2.
We now illustrate the results obtained in Theorem 1.2 through some examples and indicate possible applications of the bounds.
Example 1.4. Let
We choose the branch of the logarithm such that $\log 1=0$ . One can check that $f\in \mathcal {F}(p)$ and has the expansion
Here, $a_2=1/p-1/2$ and as a result, an application of Theorem 1.2 yields
Example 1.5. Let $f(z)=-pz\exp (z)/(z-p)$ , $z\in {\mathbb D}$ , with the expansion
Thus, $pa_2+p^2-1=p^2+p$ and $d_p(f)\leq p\exp (-(p^2+p)/2)$ .
Example 1.6 (Univalent case)
Consider $f(z)=-zp/(z-p)(1-pz)$ , $z\in {\mathbb D}$ . It is a simple exercise to check that f is one–one in ${\mathbb D}$ (see [Reference Avkhadiev and Wirths1, Reference Bhowmik2]). The Taylor expansion of this function yields the second Taylor coefficient as $a_2=p+1/p$ . Therefore, according to Theorem 1.2, we must have $d_p(f)\leq p\exp (-p^2)$ . Now for this function,
Therefore, $d_p(f)\geq p/(1+p)^2$ . Now, if $z=x$ , $x\in (-1, 0)$ , then
for all $p\in (0,1)$ . Thus, the obtained bound in Theorem 1.2 is not sharp for this univalent function.
In the above three examples, it is difficult to give the exact estimate for the distance from the origin to the image of the unit circle under f, but nonetheless, we obtain some information about this distance.
Example 1.7 (Existence of a zero)
As an application of Theorem 1.2, we wish to investigate the existence of a zero for a meromorphic function f with a nonzero pole other than at the origin. To this end, consider $p=1/2$ and the function
Suppose $f/z$ does not vanish in ${\mathbb D} \setminus \{0\}$ . Then it is clear that $f\in \mathcal {F}(1/2)$ . Expanding f in a Taylor series about the origin for $|z|<1/2$ gives
Here, $a_2=17$ . Therefore, an application of Theorem 1.2 yields
However, then we see that
This is a contradiction, and therefore $f/z$ must vanish at a nonzero point in ${\mathbb D}$ .
2 Generalisation of the main result
In this section, we generalise Theorem 1.2 by allowing the functions in $\mathcal {F}(p)$ to have more than one nonzero simple pole in ${\mathbb D}$ . This extension is possible if these poles in ${\mathbb D}$ lie on a line passing through the origin, that is, all the poles have the same argument. Thus, it will be sufficient to consider these nonzero poles in the interval $(0,1)$ as we did for one nonzero pole in ${\mathbb D}$ (see (1.3)). More precisely, we consider functions f that are meromorphic having simple poles at $z=p_1, p_2,\ldots , p_n \in (0,1)$ inside the unit disk ${\mathbb D}$ with the Taylor series expansion
where $p:=\min \, \{p_1, p_2,\ldots , p_n\}$ , ${\mathbb D}_p:=\{z\in {\mathbb C} : |z|<p\}$ and f does not vanish in ${\mathbb D}$ other than at the origin. For such f, we have $f(0)=0=f'(0)-1$ . We denote the class of such functions by $\mathcal {F}$ . Let
For $f\in \mathcal {F}$ , we define
and if $(z-p_1)(z-p_2)\ldots (z-p_n)f$ is bounded in ${\mathbb D}$ , we define
In the next theorem, we obtain estimates for $m_p(f)$ and $M_p(f)$ .
Theorem 2.1. Let $f\in \mathcal {F}$ have the expansion (2.1) in ${\mathbb D}_p$ . Then
and if $(z-p_1)(z-p_2)\ldots (z-p_n)f$ is bounded in ${\mathbb D}$ , then
These bounds are best possible.
Proof. To prove this theorem, we use a similar technique to that in the proof of Theorem 1.2. Let $s>1$ be such that
Therefore, we must have
where we choose that branch of the logarithm for which $\log f'(0)=0$ . A minor simplification of this inequality yields
For $z\in {\mathbb D}$ , we define
which is analytic in ${\mathbb D}$ by choosing that branch of logarithm for which $\log (f'(0))=0$ . By virtue of the previous inequality, we have $\mathrm {Re}\, F(z)\geq 0$ with $F(0)=1$ . Now we can expand F about the origin to get
An application of Caratheodory’s lemma for the function F in ${\mathbb D}_p$ yields
Now, letting $m_p(f)=(\prod _{i=1}^{n} p_i)/s$ , we obtain the first estimate of the theorem. To obtain the second estimate of the theorem, we let
Note that $g\in \mathcal {F}$ as $(z-p_1)(z-p_2)\ldots (z-p_n)f$ is bounded in ${\mathbb D}$ and g has the Taylor expansion
We thus have $m_p(g)/(\prod _{i=1}^{n} p_i)=(\prod _{i=1}^{n} p_i)/M_p(f)$ . Therefore, we deduce that
The above inequality follows by applying the first part of the theorem to the function g. Consequently, the second inequality of the theorem follows.
The bounds obtained in the theorem are best possible in the following sense. We consider the following functions in $\mathcal {F}$ :
where $\psi (z)=z(1-pz)/(p-z)$ . A little computation yields
Here, $a_2={\alpha }/{p}+\sum _{i=1}^{n}({1}/{p_i}-p_i)$ , which in turn implies $p|a_2+\sum _{i=1}^{n}(p_i-{1}/{p_i})|=\alpha $ and
where $\phi \in (0, \pi )\cup (\pi , 2\pi )$ . Again for the function $f^{-}_{\alpha }$ , we have
which gives $a_2=-{\alpha }/{p}+\sum _{i=1}^{n}({1}/{p_i}-p_i)$ or equivalently $p|a_2+\sum _{i=1}^{n}(p_i-{1}/{p_i})|=\alpha $ and
where $\phi \in (0, \pi )\cup (\pi , 2\pi )$ . This shows that the estimates stated in the theorem are best possible and completes the proof of the theorem.