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\title{An Extension of Wirtinger's Inequality to the Complex Integral}
\author{Silvestru Sever Dragomir}
\address{`Mathematics, College of Engineering \& Science\\
Victoria University, PO Box 14428\\
Melbourne City, MC 8001, Australia.\\
DST-NRF Centre of Excellence in the Mathematical \\
and Statistical Sciences, School of Computer Science\\
\& Applied Mathematics, University of the Witwatersrand, \\
Private Bag 3, Johannesburg 2050, South Africa}
\email{sever.dragomir@vu.edu.au}

%
\subjclass{26D15; 26D10.}
\keywords{Wirtinger's inequality, Trapezoid inequality, Complex integral,
Analytic functions.}
\begin{abstract}
In this paper we establish a natural extension of the Wirtinger inequality
to the case of complex integral of analytic functions. Applications related
to the trapezoid inequalities are also provided. Examples for logarithmic
and exponential complex functions are given as well.
\end{abstract}
\maketitle


\section{Introduction}

It is well known that, see for instance \cite{DM}, or \cite{J}, if $u\in
C^{1}([a,b],\mathbb{R})$ satisfies $u(a)=u(b)=0$, then we have the \textit{%
Wirtinger inequality} 
\begin{equation}
\int_{a}^{b}u^{2}\left( t\right) dt\leq \frac{\left( b-a\right) ^{2}}{\pi
^{2}}\int_{a}^{b}\left[ u^{\prime }\left( t\right) \right] ^{2}dt
\label{e.1.1}
\end{equation}%
with the equality holding if and only if $u\left( t\right) =K\sin \left[ 
\frac{\pi \left( t-a\right) }{b-a}\right] $ for some constant $K\in \mathbb{R%
}.$

If $u\in C^{1}([a,b],\mathbb{R})$ satisfies the condition $u(a)=0,$ then
also 
\begin{equation}
\int_{a}^{b}u^{2}\left( t\right) dt\leq \frac{4\left( b-a\right) ^{2}}{\pi
^{2}}\int_{a}^{b}\left[ u^{\prime }\left( t\right) \right] ^{2}dt
\label{e.1.2}
\end{equation}%
and the equality holds if and only if $u\left( t\right) =L\sin \left[ \frac{%
\pi \left( t-a\right) }{2\left( b-a\right) }\right] $ for some constant $%
L\in \mathbb{R}.$

If $h\in C^{1}([a,b],\mathbb{C})$ is a function with complex values and $%
h(a)=h(b)=0,$ then $\func{Re}h(a)=\func{Re}h(b)=0$ and $\func{Im}h(a)=\func{%
Im}h(b)=0$ and by writing (\ref{e.1.1}) for $\func{Re}h$ and $\func{Im}h$
and adding the obtained inequalities, we get 
\begin{equation}
\int_{a}^{b}\left\vert h\left( t\right) \right\vert ^{2}dt\leq \frac{\left(
b-a\right) ^{2}}{\pi ^{2}}\int_{a}^{b}\left\vert h^{\prime }\left( t\right)
\right\vert ^{2}dt  \label{e.1.3}
\end{equation}%
with the equality holding if and only if 
\begin{equation*}
h\left( t\right) =K\sin \left[ \frac{\pi \left( t-a\right) }{b-a}\right]
\end{equation*}%
for some complex constant $K\in \mathbb{C}.$

Similarly, if $h\in C^{1}([a,b],\mathbb{C})$ with $h(a)=0,$ then by (\ref%
{e.1.2}) we have 
\begin{equation}
\int_{a}^{b}\left\vert h\left( t\right) \right\vert ^{2}dt\leq \frac{4\left(
b-a\right) ^{2}}{\pi ^{2}}\int_{a}^{b}\left\vert h^{\prime }\left( t\right)
\right\vert ^{2}dt  \label{e.1.4}
\end{equation}%
and the equality holds if and only if 
\begin{equation*}
h\left( t\right) =L\sin \left[ \frac{\pi \left( t-a\right) }{2\left(
b-a\right) }\right]
\end{equation*}%
for some complex constant $L\in \mathbb{R}.$

For some related Wirtinger type integral inequalities see \cite{A}, \cite{B}%
, \cite{DM} and \cite{G}-\cite{R}.

In order to extend this result for the complex integral, we need some
preparations as follows.

Suppose $\gamma $ is a smooth path parametrized by $z\left( t\right) ,$ $%
t\in \left[ a,b\right] $ and $f$ is a complex function which is continuous
on $\gamma .$ Put $z\left( a\right) =u$ and $z\left( b\right) =w$ with $u,$ $%
w\in \mathbb{C}$. We define the integral of $f$ on $\gamma _{u,w}=\gamma $ as%
\begin{equation*}
\int_{\gamma }f\left( z\right) dz=\int_{\gamma _{u,w}}f\left( z\right)
dz:=\int_{a}^{b}f\left( z\left( t\right) \right) z^{\prime }\left( t\right)
dt.
\end{equation*}%
We observe that that the actual choice of parametrization of $\gamma $ does
not matter.

This definition immediately extends to paths that are piecewise smooth.
Suppose $\gamma $ is parametrized by $z\left( t\right) $, $t\in \left[ a,b%
\right] $, which is differentiable on the intervals $[a,c]$ and $[c,b],$
then assuming that $f$ is continuous on $\gamma $ we define%
\begin{equation*}
\int_{\gamma _{u,w}}f\left( z\right) dz:=\int_{\gamma _{u,v}}f\left(
z\right) dz+\int_{\gamma _{v,w}}f\left( z\right) dz
\end{equation*}%
where $v:=z\left( c\right) .$ This can be extended for a finite number of
intervals.

We also define the integral with respect to arc-length 
\begin{equation*}
\int_{\gamma _{u,w}}f\left( z\right) \left\vert dz\right\vert
:=\int_{a}^{b}f\left( z\left( t\right) \right) \left\vert z^{\prime }\left(
t\right) \right\vert dt
\end{equation*}%
and the length of the curve $\gamma $ is then 
\begin{equation*}
\ell \left( \gamma \right) =\int_{\gamma _{u,w}}\left\vert dz\right\vert
=\int_{a}^{b}\left\vert z^{\prime }\left( t\right) \right\vert dt.
\end{equation*}

Let $f$ and $g$ be holomorphic in $G$, and open domain and suppose $\gamma
\subset G$ is a piecewise smooth path from $z\left( a\right) =u$ to $z\left(
b\right) =w$. Then we have the \textit{integration by parts formula}%
\begin{equation}
\int_{\gamma _{u,w}}f\left( z\right) g^{\prime }\left( z\right) dz=f\left(
w\right) g\left( w\right) -f\left( u\right) g\left( u\right) -\int_{\gamma
_{u,w}}f^{\prime }\left( z\right) g\left( z\right) dz.  \label{e.1.5}
\end{equation}%
We recall also the \textit{triangle inequality} for the complex integral,
namely%
\begin{equation}
\left\vert \int_{\gamma }f\left( z\right) dz\right\vert \leq \int_{\gamma
}\left\vert f\left( z\right) \right\vert \left\vert dz\right\vert \leq
\left\Vert f\right\Vert _{\gamma ,\infty }\ell \left( \gamma \right)
\label{e.1.6}
\end{equation}%
where $\left\Vert f\right\Vert _{\gamma ,\infty }:=\sup_{z\in \gamma
}\left\vert f\left( z\right) \right\vert .$

We also define the $p$-norm with $p\geq 1$ by%
\begin{equation*}
\left\Vert f\right\Vert _{\gamma ,p}:=\left( \int_{\gamma }\left\vert
f\left( z\right) \right\vert ^{p}\left\vert dz\right\vert \right) ^{1/p}.
\end{equation*}%
For $p=1$ we have 
\begin{equation*}
\left\Vert f\right\Vert _{\gamma ,1}:=\int_{\gamma }\left\vert f\left(
z\right) \right\vert \left\vert dz\right\vert .
\end{equation*}%
If $p,$ $q>1$ with $\frac{1}{p}+\frac{1}{q}=1,$ then by H\"{o}lder's
inequality we have%
\begin{equation*}
\left\Vert f\right\Vert _{\gamma ,1}\leq \left[ \ell \left( \gamma \right) %
\right] ^{1/q}\left\Vert f\right\Vert _{\gamma ,p}.
\end{equation*}

In this paper we establish a natural extension of the Wirtinger inequality
to the case of complex integral of analytic functions. Applications related
to the trapezoid inequalities are also provided. Examples for logarithmic
and exponential complex functions are given as well.

\section{Wirtinger Type Inequalities}

We have the following weighted version of Wirtinger inequality:

\begin{lemma}
\label{l.2.1}Let $g:\left[ a,b\right] \rightarrow \left[ g\left( a\right)
,g\left( b\right) \right] $ be a \textit{continuous strictly increasing
function }that is \textit{of class }$C^{1}$ on $\left( a,b\right) .$

\begin{enumerate}
\item[(i)] If $h\in C^{1}([a,b],\mathbb{C})$ is a function with complex
values and $h(a)=h(b)=0,$ then%
\begin{equation}
\int_{a}^{b}\left\vert h\left( t\right) \right\vert ^{2}g^{\prime }\left(
t\right) dt\leq \frac{\left[ g\left( b\right) -g\left( a\right) \right] ^{2}%
}{\pi ^{2}}\int_{a}^{b}\frac{\left\vert h^{\prime }\left( t\right)
\right\vert ^{2}}{g^{\prime }\left( t\right) }dt.  \label{e.2.1}
\end{equation}%
The equality holds in (\ref{e.2.1}) iff 
\begin{equation*}
h\left( t\right) =K\sin \left[ \frac{\pi \left( g\left( t\right) -g\left(
a\right) \right) }{g\left( b\right) -g\left( a\right) }\right] ,\text{ }K\in 
\mathbb{C}.
\end{equation*}

\item[(ii)] If $h\in C^{1}([a,b],\mathbb{C})$ is a function with complex
values and $h(a)=0,$ then%
\begin{equation}
\int_{a}^{b}\left\vert h\left( t\right) \right\vert ^{2}g^{\prime }\left(
t\right) dt\leq \frac{4\left[ g\left( b\right) -g\left( a\right) \right] ^{2}%
}{\pi ^{2}}\int_{a}^{b}\frac{\left\vert h^{\prime }\left( t\right)
\right\vert ^{2}}{g^{\prime }\left( t\right) }dt.  \label{e.2.2}
\end{equation}%
The equality holds in (\ref{e.2.2}) iff 
\begin{equation*}
h\left( t\right) =K\sin \left[ \frac{\pi \left( g\left( t\right) -g\left(
a\right) \right) }{2\left( g\left( b\right) -g\left( a\right) \right) }%
\right] ,\text{ }K\in \mathbb{C}.
\end{equation*}
\end{enumerate}
\end{lemma}

\begin{proof}
(i) We write the inequality (\ref{e.1.3}) for the function $h=h\circ g^{-1}$
on the interval $\left[ g\left( a\right) ,g\left( b\right) \right] $ to get 
\begin{equation}
\int_{g\left( a\right) }^{g\left( b\right) }\left\vert \left( h\circ
g^{-1}\right) \left( z\right) \right\vert ^{2}dz\leq \frac{\left( g\left(
b\right) -g\left( a\right) \right) ^{2}}{\pi ^{2}}\int_{g\left( a\right)
}^{g\left( b\right) }\left\vert \left( h\circ g^{-1}\right) ^{\prime }\left(
z\right) \right\vert ^{2}dz.  \label{e.2.3}
\end{equation}%
If $h:\left[ c,d\right] \rightarrow \mathbb{C}$ is absolutely continuous on $%
\left[ c,d\right] ,$ then $h\circ g^{-1}:\left[ g\left( c\right) ,g\left(
d\right) \right] \rightarrow \mathbb{C}$ is absolutely continuous on $\left[
g\left( c\right) ,g\left( d\right) \right] $ and using the chain rule and
the derivative of inverse functions we have 
\begin{equation}
\left( h\circ g^{-1}\right) ^{\prime }\left( z\right) =\left( h^{\prime
}\circ g^{-1}\right) \left( z\right) \left( g^{-1}\right) ^{\prime }\left(
z\right) =\frac{\left( h^{\prime }\circ g^{-1}\right) \left( z\right) }{%
\left( g^{\prime }\circ g^{-1}\right) \left( z\right) }  \label{Id}
\end{equation}%
for almost every (a.e.) $z\in \left[ g\left( c\right) ,g\left( d\right) %
\right] .$

Using the inequality (\ref{e.2.3}) we then get 
\begin{equation}
\int_{g\left( a\right) }^{g\left( b\right) }\left\vert \left( h\circ
g^{-1}\right) \left( z\right) \right\vert ^{2}dz\leq \frac{\left( g\left(
b\right) -g\left( a\right) \right) ^{2}}{\pi ^{2}}\int_{g\left( a\right)
}^{g\left( b\right) }\left\vert \frac{\left( h^{\prime }\circ g^{-1}\right)
\left( z\right) }{\left( g^{\prime }\circ g^{-1}\right) \left( z\right) }%
\right\vert ^{2}dz,  \label{e.2.4}
\end{equation}%
provided $\left( h\circ g^{-1}\right) \left( g\left( a\right) \right)
=h\left( a\right) =0$ and $\left( h\circ g^{-1}\right) \left( g\left(
b\right) \right) =h\left( b\right) =0.$

Observe also that, by the change of variable $t=g^{-1}\left( z\right) ,$ $%
z\in \left[ g\left( a\right) ,g\left( b\right) \right] ,$ we have $z=g\left(
t\right) $ that gives $dz=g^{\prime }\left( t\right) dt$ and 
\begin{equation}
\int_{g\left( a\right) }^{g\left( b\right) }\left\vert \left( h\circ
g^{-1}\right) \left( z\right) \right\vert ^{2}dz=\int_{a}^{b}\left\vert
h\left( t\right) \right\vert ^{2}g^{\prime }\left( t\right) dt.
\label{e.2.9}
\end{equation}%
We also have 
\begin{equation*}
\int_{g\left( a\right) }^{g\left( b\right) }\left\vert \frac{\left(
h^{\prime }\circ g^{-1}\right) \left( z\right) }{\left( g^{\prime }\circ
g^{-1}\right) \left( z\right) }\right\vert ^{2}dz=\int_{a}^{b}\left\vert 
\frac{h^{\prime }\left( t\right) }{g^{\prime }\left( t\right) }\right\vert
^{2}g^{\prime }\left( t\right) dt=\int_{a}^{b}\frac{\left\vert h^{\prime
}\left( t\right) \right\vert ^{2}}{g^{\prime }\left( t\right) }dt.
\end{equation*}%
By making use of (\ref{e.2.4}) we get (\ref{e.2.1}).

The equality holds in (\ref{e.2.4}) provided%
\begin{equation*}
\left( h\circ g^{-1}\right) \left( z\right) =K\sin \left[ \frac{\pi \left(
z-g\left( a\right) \right) }{g\left( b\right) -g\left( a\right) }\right] ,%
\text{ }K\in \mathbb{C}
\end{equation*}%
for $z\in \left[ g\left( a\right) ,g\left( b\right) \right] .$ If we take $%
t\in \left[ a,b\right] $ and $z=g\left( t\right) ,$ we then get 
\begin{equation*}
h\left( t\right) =K\sin \left[ \frac{\pi \left( g\left( t\right) -g\left(
a\right) \right) }{g\left( b\right) -g\left( a\right) }\right] ,\text{ }K\in 
\mathbb{C}\text{.}
\end{equation*}

(ii) Follows in a similar way by (\ref{e.1.4}).
\end{proof}

If $w:\left[ a,b\right] \rightarrow \mathbb{R}$ is continuous and positive
on the interval $\left[ a,b\right] ,$ then the function $W:\left[ a,b\right]
\rightarrow \lbrack 0,\infty ),$ $W\left( x\right) :=\int_{a}^{x}w\left(
s\right) ds$ is strictly increasing and differentiable on $\left( a,b\right)
.$ We have $W^{\prime }\left( x\right) =w\left( x\right) $ for any $x\in
\left( a,b\right) .$

\begin{corollary}
\label{c.2.1}Assume that $w:\left[ a,b\right] \rightarrow \left( 0,\infty
\right) $ is continuous on $\left[ a,b\right] $ and $h\in C^{1}([a,b],%
\mathbb{C})$ is a function with complex values and $h(a)=h(b)=0,$ then%
\begin{equation}
\int_{a}^{b}\left\vert h\left( t\right) \right\vert ^{2}w\left( t\right)
dt\leq \frac{1}{\pi ^{2}}\left( \int_{a}^{b}w\left( s\right) ds\right)
^{2}\int_{a}^{b}\frac{\left\vert h^{\prime }\left( t\right) \right\vert ^{2}%
}{w\left( t\right) }dt.  \label{e.2.16}
\end{equation}%
The equality holds in (\ref{e.2.16}) iff 
\begin{equation*}
h\left( t\right) =K\sin \left[ \frac{\pi \int_{a}^{t}w\left( s\right) ds}{%
\int_{a}^{b}w\left( s\right) ds}\right] ,\text{ }K\in \mathbb{C}.
\end{equation*}

If $h(a)=0,$ then 
\begin{equation}
\int_{a}^{b}\left\vert h\left( t\right) \right\vert ^{2}w\left( t\right)
dt\leq \frac{4}{\pi ^{2}}\left( \int_{a}^{b}w\left( s\right) ds\right)
^{2}\int_{a}^{b}\frac{\left\vert h^{\prime }\left( t\right) \right\vert ^{2}%
}{w\left( t\right) }dt  \label{e.2.17}
\end{equation}%
with equality iff 
\begin{equation*}
h\left( t\right) =K\sin \left[ \frac{\pi \int_{a}^{t}w\left( s\right) ds}{%
2\int_{a}^{b}w\left( s\right) ds}\right] ,\text{ }K\in \mathbb{C}.
\end{equation*}
\end{corollary}

We have the following Wirtinger type inequality for complex functions:

\begin{theorem}
\label{t.2.1}Let $f$ be analytic in $G,$ a domain of complex numbers and
suppose $\gamma \subset G$ is a smooth path parametrized by $z\left(
t\right) ,$ $t\in \left[ a,b\right] $ from $z\left( a\right) =u$ to $z\left(
b\right) =w$ and $z^{\prime }\left( t\right) \neq 0$ for $t\in \left(
a,b\right) .$

\begin{enumerate}
\item[(i)] If $f\left( u\right) =f\left( w\right) =0$, then 
\begin{equation}
\int_{\gamma }\left\vert f\left( z\right) \right\vert ^{2}\left\vert
dz\right\vert \leq \frac{1}{\pi ^{2}}\ell ^{2}\left( \gamma \right)
\int_{\gamma }\left\vert f^{\prime }\left( z\right) \right\vert
^{2}\left\vert dz\right\vert .  \label{e.2.18}
\end{equation}%
The equality holds in (\ref{e.2.18}) iff 
\begin{equation}
f\left( v\right) =K\sin \left[ \frac{\pi \ell \left( \gamma _{u,v}\right) }{%
\ell \left( \gamma \right) }\right] ,\text{ }K\in \mathbb{C}
\label{e.2.18.a}
\end{equation}%
where $v=z\left( t\right) ,$ $t\in \left[ a,b\right] $ and $\ell \left(
\gamma _{u,v}\right) =\int_{a}^{t}\left\vert z^{\prime }\left( s\right)
\right\vert ds.$

\item[(ii)] If $f\left( u\right) =0$, then%
\begin{equation}
\int_{\gamma }\left\vert f\left( z\right) \right\vert ^{2}\left\vert
dz\right\vert \leq \frac{4}{\pi ^{2}}\ell ^{2}\left( \gamma \right)
\int_{\gamma }\left\vert f^{\prime }\left( z\right) \right\vert
^{2}\left\vert dz\right\vert .  \label{e.2.19}
\end{equation}%
The equality holds in (\ref{e.2.19}) iff 
\begin{equation}
f\left( v\right) =K\sin \left[ \frac{\pi \ell \left( \gamma _{u,v}\right) }{%
2\ell \left( \gamma \right) }\right] ,\text{ }K\in \mathbb{C}
\label{e.2.19.a}
\end{equation}%
where $v=z\left( t\right) ,$ $t\in \left[ a,b\right] $.
\end{enumerate}
\end{theorem}

\begin{proof}
(i) Consider the function $h\left( t\right) =f\left( z\left( t\right)
\right) $ and $w\left( t\right) =\left\vert z^{\prime }\left( t\right)
\right\vert ,$ $t\in \left[ a,b\right] .$ Then $h^{\prime }\left( t\right)
=\left( f\left( z\left( t\right) \right) \right) ^{\prime }=f^{\prime
}\left( z\left( t\right) \right) z^{\prime }\left( t\right) $ for $t\in
\left( a,b\right) .$ Also $h\left( a\right) =f\left( z\left( a\right)
\right) =f\left( u\right) =0$ and $h\left( b\right) =f\left( z\left(
b\right) \right) =f\left( w\right) =0.$ By utilising the inequality (\ref%
{e.2.16}) for these choices, we get 
\begin{align*}
\int_{a}^{b}\left\vert f\left( z\left( t\right) \right) \right\vert
^{2}\left\vert z^{\prime }\left( t\right) \right\vert dt& \leq \frac{1}{\pi
^{2}}\left( \int_{a}^{b}\left\vert z^{\prime }\left( s\right) \right\vert
ds\right) ^{2}\int_{a}^{b}\frac{\left\vert f^{\prime }\left( z\left(
t\right) \right) z^{\prime }\left( t\right) \right\vert ^{2}}{\left\vert
z^{\prime }\left( t\right) \right\vert }dt \\
& =\frac{1}{\pi ^{2}}\left( \int_{a}^{b}\left\vert z^{\prime }\left(
s\right) \right\vert ds\right) ^{2}\int_{a}^{b}\frac{\left\vert f^{\prime
}\left( z\left( t\right) \right) \right\vert ^{2}\left\vert z^{\prime
}\left( t\right) \right\vert ^{2}}{\left\vert z^{\prime }\left( t\right)
\right\vert }dt \\
& =\frac{1}{\pi ^{2}}\left( \int_{a}^{b}\left\vert z^{\prime }\left(
s\right) \right\vert ds\right) ^{2}\int_{a}^{b}\left\vert f^{\prime }\left(
z\left( t\right) \right) \right\vert ^{2}\left\vert z^{\prime }\left(
t\right) \right\vert dt,
\end{align*}%
which is equivalent to (\ref{e.2.18}).

The equality (\ref{e.2.18.a}) follows by the corresponding equality in
Corollary \ref{c.2.1}.

(ii) Follows by the corresponding result from Corollary \ref{c.2.1}.
\end{proof}

\section{Some Trapezoid Type Inequalities}

We have:

\begin{proposition}
\label{p.3.1}Let $g$ be analytic in $G,$ a domain of complex numbers and
suppose $\gamma \subset G$ is a smooth path parametrized by $z\left(
t\right) ,$ $t\in \left[ a,b\right] $ from $z\left( a\right) =u$ to $z\left(
b\right) =w,$ $w\neq u$ and $z^{\prime }\left( t\right) \neq 0$ for $t\in
\left( a,b\right) .$ Then%
\begin{multline}
\left\vert \frac{1}{w-u}\int_{\gamma }g\left( z\right) dz-\frac{g\left(
u\right) +g\left( w\right) }{2}\right\vert  \label{e.3.1} \\
\leq \frac{1}{\pi }\frac{\ell \left( \gamma \right) }{\left\vert
w-u\right\vert }\left( \frac{1}{\ell \left( \gamma \right) }\int_{\gamma
}\left\vert g^{\prime }\left( z\right) -\frac{g\left( w\right) -g\left(
u\right) }{w-u}\right\vert ^{2}\left\vert dz\right\vert \right) ^{1/2}.
\end{multline}
\end{proposition}

\begin{proof}
Consider the function $f:G\rightarrow \mathbb{C}$ defined by 
\begin{equation*}
f\left( z\right) :=g\left( z\right) -\frac{g\left( u\right) \left(
w-z\right) +g\left( w\right) \left( z-u\right) }{w-u},\text{ }z\in \gamma .
\end{equation*}%
Observe that $f\left( u\right) =f\left( w\right) =0$ and by (\ref{e.2.18})
we get 
\begin{multline}
\int_{\gamma }\left\vert g\left( z\right) -\frac{g\left( u\right) \left(
w-z\right) +g\left( w\right) \left( z-u\right) }{w-u}\right\vert
^{2}\left\vert dz\right\vert  \label{e.3.2} \\
\leq \frac{1}{\pi ^{2}}\ell ^{2}\left( \gamma \right) \int_{\gamma
}\left\vert g^{\prime }\left( z\right) -\frac{g\left( w\right) -g\left(
u\right) }{w-u}\right\vert ^{2}\left\vert dz\right\vert .
\end{multline}%
Using Cauchy-Bunyakovsky-Schwarz integral inequality we also have%
\begin{multline*}
\left\vert \int_{\gamma }\left[ g\left( z\right) -\frac{g\left( u\right)
\left( w-z\right) +g\left( w\right) \left( z-u\right) }{w-u}\right]
dz\right\vert ^{2} \\
\leq \int_{\gamma }\left\vert dz\right\vert \int_{\gamma }\left\vert g\left(
z\right) -\frac{g\left( u\right) \left( w-z\right) +g\left( w\right) \left(
z-u\right) }{w-u}\right\vert ^{2}\left\vert dz\right\vert \\
=\ell \left( \gamma \right) \int_{\gamma }\left\vert g\left( z\right) -\frac{%
g\left( u\right) \left( w-z\right) +g\left( w\right) \left( z-u\right) }{w-u}%
\right\vert ^{2}\left\vert dz\right\vert
\end{multline*}%
and since 
\begin{multline*}
\int_{\gamma }\left[ g\left( z\right) -\frac{g\left( u\right) \left(
w-z\right) +g\left( w\right) \left( z-u\right) }{w-u}\right] dz \\
=\int_{\gamma }g\left( z\right) dz-\frac{g\left( u\right) \int_{\gamma
}\left( w-z\right) dz+g\left( w\right) \int_{\gamma }\left( z-u\right) dz}{%
w-u} \\
=\int_{\gamma }g\left( z\right) dz-\frac{g\left( u\right) +g\left( w\right) 
}{2}\left( w-u\right) ,
\end{multline*}%
hence%
\begin{multline*}
\left\vert \int_{\gamma }g\left( z\right) dz-\frac{g\left( u\right) +g\left(
w\right) }{2}\left( w-u\right) \right\vert ^{2} \\
\leq \ell \left( \gamma \right) \int_{\gamma }\left\vert g\left( z\right) -%
\frac{g\left( u\right) \left( w-z\right) +g\left( w\right) \left( z-u\right) 
}{w-u}\right\vert ^{2}\left\vert dz\right\vert
\end{multline*}%
and by (\ref{e.3.2}) we get%
\begin{multline*}
\left\vert \int_{\gamma }g\left( z\right) dz-\frac{g\left( u\right) +g\left(
w\right) }{2}\left( w-u\right) \right\vert ^{2} \\
\leq \ell \left( \gamma \right) \int_{\gamma }\left\vert g\left( z\right) -%
\frac{g\left( u\right) \left( w-z\right) +g\left( w\right) \left( z-u\right) 
}{w-u}\right\vert ^{2}\left\vert dz\right\vert \\
\leq \frac{1}{\pi ^{2}}\ell ^{3}\left( \gamma \right) \int_{\gamma
}\left\vert g^{\prime }\left( z\right) -\frac{g\left( w\right) -g\left(
u\right) }{w-u}\right\vert ^{2}\left\vert dz\right\vert ,
\end{multline*}%
which implies the desired result (\ref{e.3.1}).
\end{proof}

We also have:

\begin{proposition}
\label{p.3.2}Let $g$ be analytic in $G,$ a domain of complex numbers and
suppose $\gamma \subset G$ is a smooth path parametrized by $z\left(
t\right) ,$ $t\in \left[ a,b\right] $ from $z\left( a\right) =u$ to $z\left(
b\right) =w,$ $w\neq u$ and $z^{\prime }\left( t\right) \neq 0$ for $t\in
\left( a,b\right) .$ If $u+w-z\in G$ for $z\in \gamma ,$ then%
\begin{multline}
\left\vert \frac{1}{w-u}\int_{\gamma }\widehat{g\left( z\right) }dz-\frac{%
g\left( u\right) +g\left( w\right) }{2}\right\vert  \label{e.3.3} \\
\leq \frac{1}{2\pi }\frac{\ell \left( \gamma \right) }{\left\vert
w-u\right\vert }\left( \frac{1}{\ell \left( \gamma \right) }\int_{\gamma
}\left\vert g^{\prime }\left( z\right) -g^{\prime }\left( u+w-z\right)
\right\vert ^{2}\left\vert dz\right\vert \right) ^{1/2},
\end{multline}%
where 
\begin{equation*}
\widehat{g\left( z\right) }:=\frac{g\left( z\right) +g\left( u+w-z\right) }{2%
},\text{ }z\in \gamma .
\end{equation*}
\end{proposition}

\begin{proof}
Consider the function $f:G\rightarrow \mathbb{C}$ defined by%
\begin{equation*}
f\left( z\right) :=\frac{g\left( z\right) +g\left( u+w-z\right) }{2}-\frac{%
g\left( u\right) +g\left( w\right) }{2},\text{ }z\in \gamma .
\end{equation*}%
Observe that $f\left( u\right) =f\left( w\right) =0$ and by (\ref{e.2.18})
we get 
\begin{multline}
\int_{\gamma }\left\vert \frac{g\left( z\right) +g\left( u+w-z\right) }{2}-%
\frac{g\left( u\right) +g\left( w\right) }{2}\right\vert ^{2}\left\vert
dz\right\vert  \label{e.3.4} \\
\leq \frac{1}{4\pi ^{2}}\ell ^{2}\left( \gamma \right) \int_{\gamma
}\left\vert g^{\prime }\left( z\right) -g^{\prime }\left( u+w-z\right)
\right\vert ^{2}\left\vert dz\right\vert .
\end{multline}

By Cauchy-Bunyakovsky-Schwarz integral inequality we have%
\begin{multline*}
\left\vert \int_{\gamma }\left[ \frac{g\left( z\right) +g\left( u+w-z\right) 
}{2}-\frac{g\left( u\right) +g\left( w\right) }{2}\right] dz\right\vert ^{2}
\\
\leq \int_{\gamma }\left\vert dz\right\vert \int_{\gamma }\left\vert \frac{%
g\left( z\right) +g\left( u+w-z\right) }{2}-\frac{g\left( u\right) +g\left(
w\right) }{2}\right\vert ^{2}\left\vert dz\right\vert ,
\end{multline*}%
which gives%
\begin{multline*}
\left\vert \int_{\gamma }\frac{g\left( z\right) +g\left( u+w-z\right) }{2}dz-%
\frac{g\left( u\right) +g\left( w\right) }{2}\left( w-u\right) \right\vert
^{2} \\
\leq \ell \left( \gamma \right) \int_{\gamma }\left\vert \frac{g\left(
z\right) +g\left( u+w-z\right) }{2}-\frac{g\left( u\right) +g\left( w\right) 
}{2}\right\vert ^{2}\left\vert dz\right\vert .
\end{multline*}%
By (\ref{e.3.4}) we then get 
\begin{align*}
& \left\vert \int_{\gamma }\frac{g\left( z\right) +g\left( u+w-z\right) }{2}%
dz-\frac{g\left( u\right) +g\left( w\right) }{2}\left( w-u\right)
\right\vert ^{2} \\
& \leq \frac{1}{4\pi ^{2}}\ell ^{3}\left( \gamma \right) \int_{\gamma
}\left\vert g^{\prime }\left( z\right) -g^{\prime }\left( u+w-z\right)
\right\vert ^{2}\left\vert dz\right\vert ,
\end{align*}%
which is equivalent to (\ref{e.3.3}).
\end{proof}

\begin{corollary}
\label{c.2.2}With the assumption of Proposition \ref{p.3.2} and if 
\begin{equation*}
\left\vert g^{\prime }\left( z\right) -g^{\prime }\left( u+w-z\right)
\right\vert \leq \left\vert 2z-u-w\right\vert L
\end{equation*}%
for some $L>0$ and for any $z\in \gamma ,$ then 
\begin{multline}
\left\vert \frac{1}{w-u}\int_{\gamma }\widehat{g\left( z\right) }dz-\frac{%
g\left( u\right) +g\left( w\right) }{2}\right\vert  \label{e.3.5} \\
\leq \frac{1}{\pi }\frac{\ell \left( \gamma \right) L}{\left\vert
w-u\right\vert }\left( \frac{1}{\ell \left( \gamma \right) }\int_{\gamma
}\left\vert z-\frac{u+w}{2}\right\vert ^{2}\left\vert dz\right\vert \right)
^{1/2}.
\end{multline}
\end{corollary}

\begin{remark}
\label{r.2.1}If $\left\Vert g^{\prime \prime }\right\Vert _{\infty
,G}:=\sup_{z\in G}\left\vert g^{\prime \prime }\left( z\right) \right\vert
<\infty ,$ then we can take above $L=\left\Vert g^{\prime \prime
}\right\Vert _{\infty ,G}$.
\end{remark}

\section{Some Examples for Logarithmic and Exponential Functions}

Consider the function $g\left( z\right) =\frac{1}{z},$ $z\in \mathbb{%
C\setminus }\left\{ 0\right\} $. Then 
\begin{equation*}
g^{\left( k\right) }\left( z\right) =\frac{\left( -1\right) ^{k}k!}{z^{k+1}}%
\text{ for }k\geq 0,\text{ }z\in \mathbb{C\setminus }\left\{ 0\right\}
\end{equation*}%
and suppose $\gamma \subset \mathbb{C}_{\ell }:=\mathbb{C\setminus }\left\{
x+iy:x\leq 0,\text{ }y=0\right\} $ is a \textit{smooth path} parametrized by 
$z\left( t\right) ,$ $t\in \left[ a,b\right] $ with $z\left( a\right) =u$
and $z\left( b\right) =w$ where $u,$ $w\in \mathbb{C}_{\ell }$ and $%
z^{\prime }\left( t\right) \neq 0$ for $t\in \left( a,b\right) .$ Then 
\begin{equation*}
\int_{\gamma }g\left( z\right) dz=\int_{\gamma _{u,w}}g\left( z\right)
dz=\int_{\gamma _{u,w}}\frac{dz}{z}=\limfunc{Log}\left( w\right) -\limfunc{%
Log}\left( u\right)
\end{equation*}%
for $u,$ $w\in \mathbb{C}_{\ell }$.

By making use of the inequality (\ref{e.3.1}) we get 
\begin{multline}
\left\vert \frac{\limfunc{Log}\left( w\right) -\limfunc{Log}\left( u\right) 
}{w-u}-\frac{u+w}{2uw}\right\vert  \label{e.4.1} \\
\leq \frac{1}{\pi }\frac{\ell \left( \gamma \right) }{\left\vert
w-u\right\vert \left\vert wu\right\vert }\left( \frac{1}{\ell \left( \gamma
\right) }\int_{\gamma }\left\vert \frac{z^{2}-uw}{z^{2}}\right\vert
^{2}\left\vert dz\right\vert \right) ^{1/2}.
\end{multline}

Observe also that 
\begin{equation*}
\int_{\gamma _{u,w}}\frac{dz}{u+w-z}=\left. -\limfunc{Log}\left(
u+w-z\right) \right\vert _{u}^{w}=-\limfunc{Log}\left( u\right) +\limfunc{Log%
}\left( w\right) ,
\end{equation*}%
therefore, by the inequality (\ref{e.3.3}) we get 
\begin{multline}
\left\vert \frac{\limfunc{Log}\left( w\right) -\limfunc{Log}\left( u\right) 
}{w-u}-\frac{u+w}{2uw}\right\vert  \label{e.4.2} \\
\leq \frac{1}{\pi }\frac{\left\vert w+u\right\vert \ell \left( \gamma
\right) }{\left\vert w-u\right\vert }\left( \frac{1}{\ell \left( \gamma
\right) }\int_{\gamma }\left\vert \frac{z-\frac{u+w}{2}}{z^{2}\left(
u+w-z\right) ^{2}}\right\vert ^{2}\left\vert dz\right\vert \right) ^{1/2}.
\end{multline}

Consider the function $g\left( z\right) =\exp z,$ $z\in \mathbb{C}$. Suppose 
$\gamma $ is a \textit{smooth path} parametrized by $z\left( t\right) ,$ $%
t\in \left[ a,b\right] $ with $z\left( a\right) =u$ and $z\left( b\right) =w$
where $u,$ $w\in \mathbb{C}$ and $z^{\prime }\left( t\right) \neq 0$ for $%
t\in \left( a,b\right) .$

By making use of the inequality (\ref{e.3.1}) we get%
\begin{multline}
\left\vert \frac{\exp w-\exp u}{w-u}-\frac{\exp u+\exp w}{2}\right\vert
\label{e.4.3} \\
\leq \frac{1}{\pi }\frac{\ell \left( \gamma \right) }{\left\vert
w-u\right\vert }\left( \frac{1}{\ell \left( \gamma \right) }\int_{\gamma
}\left\vert \exp z-\frac{\exp w-\exp u}{w-u}\right\vert ^{2}\left\vert
dz\right\vert \right) ^{1/2},
\end{multline}%
while from (\ref{e.3.3}) we get 
\begin{multline}
\left\vert \frac{\exp w-\exp u}{w-u}-\frac{\exp u+\exp w}{2}\right\vert
\label{e.4.4} \\
\leq \frac{1}{2\pi }\frac{\ell \left( \gamma \right) }{\left\vert
w-u\right\vert }\left( \frac{1}{\ell \left( \gamma \right) }\int_{\gamma
}\left\vert \exp z-\exp \left( u+w-z\right) \right\vert ^{2}\left\vert
dz\right\vert \right) ^{1/2}.
\end{multline}

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\end{document}

