\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 184, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/184\hfil Cauchy type generalizations]
{Cauchy type generalizations of holomorphic mean value theorems}

\author[ A. N. Mohapatra\hfil EJDE-2012/184\hfilneg]
{Anugraha Nidhi Mohapatra}  % in alphabetical order

\address{Anugraha Nidhi Mohapatra \newline
Department of Mathematics, Goa University, Goa, 403206, India}
\email{anm@unigoa.ac.in}

\thanks{Submitted July 13, 2012. Published October 28, 2012.}
\subjclass[2000]{26A24}
\keywords{Mean Value theorem; differentiability; holomorphic functions}

\begin{abstract}
 We extend the results on the Mean Value Theorem obtained by  
 Flett, Myers, Sahoo, Cakmak and Tiryaki to holomorphic functions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this note, we extend upon some variants of mean value theorems in
the real variable case to holomorphic function in the spirit of
Evard and Jaffari \cite{EJ}.
It is well known that the natural extension of
mean value theorems to the case of holomorphic case do not hold, for
example Rolle's theorem does not hold for $f(z) = e^z-1$, for
$z \in [0,2\pi i]$ (line joining $0$ and $2\pi i$).  Our main inputs here
are Cauchy type extensions of Flett's theorem and Myers's theorem
for a pair of functions in the real variable case.

We note that Flett's theorem \cite{F,SD} has received  a lot of attention
recently 
(see for example \cite{pw,pow2,mol,RS,MP}), where as, its twin, the Myers's theorem (see
\cite{M}) did not get that much. One possible explanation could be
that if a result holds in reference to one end point of an interval,
it is natural to guess that an identical (or similar) result holds
in reference to the other end point. However, in this note, we use
these two results side by side. We note that these variants of mean
value theorems have found applications in solving functional
equations (see for example \cite{SD}).

This is a continuation of  work in \cite{Mo}. In this note, instead
of a single function of a single variable, we consider a pair of
functions of a single variable (in both real and complex variables).

In the second section we state and prove the basic results that form
 input for the main results in Section 3. In the third section we
also prove a Cauchy type extension of the standard Cauchy mean value
theorem. Extensions of results of  both Davit et al,  and
Cakmak-Tiryaki \cite{CT} are also obtained there for a pair of holomorphic
functions. All these extensions are  in the spirit of
Evarad-Jaffari.

Throughout this note we denote by $[a,b]$ the line segment joining
$a$ and $b$ (both endpoints included) in the appropriate space
($\mathbb{R}$ or $\mathbb{C}$). In the same spirit we define
$(a,b)$, $[a,b)$, $(a,b]$. For a complex valued function $f$, its
real and imaginary parts are denoted by $\operatorname{Re}(f)$ and
$\operatorname{Im}(f)$ respectively.


\section{Basic results}
We start with Flett's theorem \cite{F,SD}.

\begin{theorem}[\cite{F}] \label{flett}
 If $f:[a,b] \to \mathbb{R}$ is differentiable on $[a,b]$ and that
 $f'(a)=f'(b)$, then there exists a $c \in (a,b)$ such that
 $$  f(c)-f(a) = (c - a) f'(c). $$
 \end{theorem}

The next one is a slight modification of the above result, known as
Myers's theorem \cite{M}.

\begin{theorem}[\cite{M}] \label{myer}
 If $f:[a,b] \to \mathbb{R}$ is differentiable on $[a,b]$ and that
 $f'(a)=f'(b)$, then there exists a $c \in (a,b)$ such that
 $$  f(b)-f(c) = (b - c) f'(c). $$
 \end{theorem}

 However, the conditions $f'(a)= f'(b)$ can be dropped to obtain
 more general results: the first one in the below is a
 generalization of Flett's result and is due to  Sahoo and Riedel
and the next one is a generalisation of Myers's result,  which we call
 Cakmak-Tiryaki theorem.

\begin{theorem}[\cite{SD}] \label{sahooreidel}
 Let $f:[a,b] \to \mathbb{R}$ be differentiable on $[a,b]$.
 Then  there exists a $c \in (a,b)$ such that
$$
f(a)-f(c) = (a - c) f'(c)+ \frac{1}{2} \frac{f'(b)-f'(a)}{b-a}(c-a)^2.
$$
\end{theorem}


\begin{theorem}[\cite{CT}] \label{cakmak}
 Let $f:[a,b] \to \mathbb{R}$ be differentiable on $[a,b]$.
 Then there exists a $c \in (a,b)$ such that
 $$
f(b)-f(c) = (b - c) f'(c)+\frac{1}{2} \frac{f'(b)-f'(a)}{b-a}(b-c)^2.
$$
\end{theorem}

Now we are ready to prove Cauchy type generalization for a pair of
functions, which is an extension of Sahoo-Riedel theorem.

\begin{theorem}\label{srgeneral}
 If $f, h:[a,b] \to \mathbb{R}$ are two
differentiable functions on $[a,b]$,
  then there exists a $c \in (a,b)$ such that
\begin{equation}\label{flett-extension}
\begin{split}
&[h(b)-h(a)] h'(b) \{ f(c) - f(a) - (c-a)f'(c) \}
 \\
&= [f'(b)-f'(a)] [h(c) - h(a)] \big\{ \frac{1}{2} [h(c)-h(a)] -
h'(c)(c-a)\big\}
\end{split}
\end{equation}
\end{theorem}

\begin{proof}
 Let
$$
g(x) = [h(b)-h(a)]f(x) h'(b)-\frac{1}{2}[f'(b) -f'(a)][h(x)-h(a)]^2.
$$
Then
$$
g'(x)= [h(b)-h(a)]f'(x)h'(b)- [f'(b)-f'(a)] [h(x)-h(b)] h'(x).
$$
Now it is easy to check that
$$
g'(a) = g'(b) = [h(b)-h(a)]f'(a)h'(b).
$$
So applying Flett's theorem \ref{flett} for $g$ and substituting the
expression of $g$ in terms of $f$ and $h$ we get the asserted result.
\end{proof}

The next result is an extension of Cakmak-Tiryaki Theorem for a pair of
functions.

\begin{theorem} \label{ctgeneral}
 If $f, h:[a,b] \to \mathbb{R}$ are two differentiable functions on $[a,b]$,
then there exists a $c \in (a,b)$ such that
\begin{equation} \label{myer-extension}
\begin{split}
&[h(b)-h(a)] h'(a) \{ f(b) - f(c) - (b-c)f'(c)  \} \\
&= [f'(b)-f'(a)] [h(c) - h(b)] \big\{ \frac{1}{2} [h(b)-h(c)] -
h'(c)(b-c)\big\}
\end{split}
\end{equation}
\end{theorem}

\begin{proof}
 Let
$$
g(x) = [h(b)-h(a)]f(x) h'(a)-\frac{1}{2}[f'(b) - f'(a)][h(x)-h(b)]^2.
$$
Then
$$
g'(x)= [h(b)-h(a)f'(x)h'(a)- [f'(b)-f'(a)] [h(x)-h(b)] h'(x).
$$
Now it is easy to check that
$$
 g'(a) = g'(b) = [h(b)-h(a)]f'(b)h'(a).
$$
So applying Myers's theorem \ref{myer} for $g$ and substituting the
expression of $g$ in terms of $f$ and $h$ we get the asserted
result.
\end{proof}


\begin{remark} \label{rmk2.7} \rm
 By  setting  $h(x)=x$ in the Theorem \ref{srgeneral}
(Theorem \ref{ctgeneral} respectively), we obtain the assertions of
Theorem \ref{sahooreidel}  (Theorem \ref{cakmak} respectively).
\end{remark}

\section{Mean Value Theorem for holomorphic functions}

In this section, in the spirit of Evard-Jaffari (see \cite{EJ}),   we will prove
some mean value theorems for holomorphic functions, which are
extensions of results of Davitt et al. (see \cite{DPRS}),
and that of Cakmak-Tiryaki.
We need the following Rolle's type  result on holomorphic functions
due to Evard and Jafari,   to prove a complex version of Cauchy type
mean value theorem.

\begin{theorem}[\cite{EJ,SD}] \label{rolle}
Let $f$ be holomorphic on a convex open domain $D$ of
$\mathbb{C}$. Let $a,b \in D$ with $a \neq b$ such that
$f(a)=f(b)=0$. Then there exists $z_1,z_2\in (a,b)$ such that
$\operatorname{Re}f'(z_1)= 0= \operatorname{Im}f'(z_2)$.
\end{theorem}

First we prove a Cauchy type of result for a pair of holomorphic
functions.

\begin{theorem} \label{thm3.2}
Let $f$ and $h$ be holomorphic on a convex open domain $D
\subset \mathbb{C}$. Let $a,b\in D$ be such that $a \neq b$. Then
there exists $z_1, z_2 \in D$ such that
\begin{equation}\label{eqn1}
 \operatorname{Re} \{f'(z_1)[h(b)-h(a)]\}  = \operatorname{Re}
\{h'(z_1)[f(b)-f(a)]\}
\end{equation}
and
\begin{equation}
\label{eqn2} \operatorname{Im} \{f'(z_2)[h(b)-h(a)]\}  = \operatorname{Im}
\{h'(z_2)[f(b)-f(a)]\}
\end{equation}
\end{theorem}

\begin{proof}
Let
$$ 
g(z)= [f(z)-f(a)][h(b)-f(a)] - [h(z)-h(a)][f(b)-f(a)]. 
$$
Then $g'(z) = f'(z)[h(b)-h(a)] - h'(z)[f(b)-f(a)]$. Since
 $g(a) = g(b)=0$, from the above theorem we conclude that there exists
$z_1,z_2 \in (a,b)$ such that 
$\operatorname{Re} g'(z_1)= 0= \operatorname{Im} g'(z_2)$, which upon expanding
 yields the identities \eqref{eqn1}
and \eqref{eqn2}.
\end{proof}

The following two theorems are holomorphic versions of Theorems
\ref{sahooreidel} and \ref{cakmak}. Our next objective is to extend
these theorems for a pair of  the holomorphic functions.
Here we use the following notation: for any $z, \omega \in
\mathbb{C}$, we denote $\langle z, \omega \rangle$ by
\begin{equation}
\label{pseudo}
 \langle z,\omega \rangle = \operatorname{Re}(z\overline{\omega})
= \operatorname{Re}(z)\operatorname{Re}(\omega)
+ \operatorname{Im}(z) \operatorname{Im}(\omega).
\end{equation}

\begin{theorem}[\cite{DPRS}] \label{sahootheorem2} 
Let $f$ be a holomorphic function defined on
an open convex subset $D$ of  $\mathbb{C}$. Let $a$ and $b$ be two
distinct points in $D$. Then there exists $z_1,z_2 \in D$ such that,
in accordance with \eqref{pseudo},
\begin{equation}
\operatorname{Re}( f'(z_1)) = \frac{\langle b-a, f(z_1) -f(a)\rangle}{\langle
b-a,z_1-a\rangle} + \frac{1}{2} \frac{{\rm Re }(f'(b)-f'(a))}{b-a}
(z_1-a)
\end{equation}
and
\begin{equation}
\operatorname{Im} (f'(z_2)) = \frac{\langle b-a, -i[f(z_2)
-f(a)]\rangle}{\langle b-a,z_2-a\rangle} + \frac{1}{2} \frac{{\rm Im
}(f'(b)-f'(a))}{b-a} (z_2-a)
\end{equation}
\end{theorem}

\begin{theorem}[\cite{CT}]\label{cttheorem2} 
Let $f$ be a holomorphic function defined on an
open convex subset $D$ of $\mathbb{C}$. Let $a$ and $b$ be distinct
points of $D$. Then there exists $z_1,z_2 \in D$ such that, in
accordance with \eqref{pseudo},
\begin{equation}
\operatorname{Re}( f'(z_1)) = \frac{\langle b-a, f(z_1) -f(b)\rangle}{\langle
b-a,z_1-b\rangle} + \frac{1}{2} \frac{{\rm Re }(f'(b)-f'(a))}{b-a}
(z_1-b)
\end{equation}
and
\begin{equation}
\operatorname{Im} (f'(z_2)) = \frac{\langle b-a, -i[f(z_2)
-f(b)]\rangle}{\langle b-a,z_2-b\rangle} + \frac{1}{2} \frac{{\rm Im
}(f'(b)-f'(a))}{b-a} (z_2-b)
\end{equation}
\end{theorem}

\begin{theorem} \label{cauchyholomorphic} 
Let $f$ and $h$ be holomorphic on a convex open domain $D \subset \mathbb{C}$. 
 Then there exist $z_1 , z_2\in (a,b)$ such that
(i)
\begin{equation}  \label{myerholomorphic}
\begin{split}
&[\operatorname{Re}(h'(a))]  \langle b-a,
h(b)-h(a)\rangle \big\{ \frac{\langle b-a,
f(b)-f(z_1)\rangle} {\langle b-a,b-z_1\rangle }
-\operatorname{Re}(f'(z_1))\big\} \\
&= \{\operatorname{Re}[f'(b)-f'(a)]\}  \langle  b-a, h(z_1)-h(b)\rangle\\
&\quad\times \big\{ \frac{1}{2}\frac{\langle b-a,
h(b)-h(z_1)\rangle}
  {\langle b-a,b-z_1\rangle } -\operatorname{Re}(h'(z_1))\big\}
\end{split}
\end{equation}
and (ii)
\begin{equation} \label{flettholomorphic1}
\begin{split}
&[\operatorname{Re}(h'(b)) \langle b-a,
h(b)-h(a)\rangle] \big\{ \frac{\langle b-a,
f(a)-f(z_2)\rangle}{\langle b-a,a-z_2\rangle }
-\operatorname{Re}f'(z_2)\big\} \\
&= \{\operatorname{Re}[f'(b)-f'(a)]\} \langle  b-a, h(z_2)-h(a)\rangle\\
&\quad\times  \big \{ \frac{1}{2}\frac{\langle b-a, h(a)-h(z_2)\rangle}
  {\langle b-a,a-z_2\rangle } -\operatorname{Re}(h'(z_2))\big\}
\end{split}
\end{equation}
\end{theorem}

\begin{proof}
 Let $\operatorname{Re}(a)=a_1$, $\operatorname{Im}(a)=a_2$,
 $\operatorname{Re}(b)=b_1$, $\operatorname{Im}(b)=b_2$,
$\operatorname{Re}(f)=u$, $\operatorname{Im}(f)=v$, $\operatorname{Re}(h)=u_1$,
 $\operatorname{Im}(h)=v_1$. For $t \in [0,1]$, define
\begin{gather*}
\phi(t) = (b_1-a_1)u(a+t(b-a)) + (b_2-a_2)v(a+t(b-a)),\\
\psi(t) = (b_1-a_1)u_1(a+t(b-a)) + (b_2-a_2)v_1(a+t(b-a)).
\end{gather*}
Then by Theorem \ref{ctgeneral} there exists a $c \in (0,1)$ such
that
\begin{equation} \label{cauchy1}
\begin{split}
& [\psi (1)-\psi(0)] \psi'(0)[\phi(1) -\phi(c)-(1-c)\phi'(c)] \\
&= [\phi'(1)-\phi'(0)][\psi(c)-\psi(1)]\big\{
\frac{1}{2}(\psi(1)-\psi(c)) - (1-c)\psi'(c) \big\}
\end{split}
\end{equation}


By taking $z=a+t(b-a)$,  $z_1 = a+c(b-a)$ and  using \eqref{pseudo},
we note that  the functions $\phi$ and $\psi$ satisfy the following
properties:
\begin{gather*}
\phi(1) = \langle b-a,f(b)\rangle, \quad
\psi(1)  = \langle b-a,h(b)\rangle,\\
\phi(0)  = \langle b-a,f(a)\rangle, \quad
\psi(0) = \langle b-a, h(a)\rangle,\\
\phi'(t) = |b-a|^2\operatorname{Re}(f'(z)), \quad
\psi'(t) = |b-a|^2{\rm Re } (h'(z)),\\
\psi'(0) = |b-a|^2\operatorname{Re}(h'(a)),\quad
\phi'(1)  = |b-a|^2\operatorname{Re}(f'(b)),\\
\phi'(0) = |b-a|^2\operatorname{Re}(f'(a)), \quad
\psi(1)-\psi(0) = \langle b-a, h(b)-h(a)\rangle,\\
\phi'(1) - \phi'(0) =  |b-a|^2 \operatorname{Re} [f'(b)-f'(a)],\\\
(1-c)\phi'(c) =  \langle b-a,b-z_1\rangle \operatorname{Re}(f'(z_1)),\\
(1-c)\psi'(c) =\langle b-a,b-z_1\rangle \operatorname{Re}(h'(z_1)),\quad
c\phi'(c) =  \langle b-a,z_1-a\rangle \operatorname{Re}(f'(z_1)),\\
c \psi'(c) =\langle b-a,z_1-a\rangle \operatorname{Re}(h'(z_1)), \quad
\psi(1)-\psi(c) =\langle b-a, h(b)-h(z_1)\rangle.
\end{gather*}
Upon substituting these expressions in
\eqref{cauchy1} yields \eqref{myerholomorphic}.


Similarly,  applying Theorem \ref{srgeneral}  for the pair of
functions $\phi$ and $\psi$ there exists a $c_1 \in (0,1)$ such that
\begin{align*}
& [\psi(1)-\psi(0)]\psi'(1) [\phi(c_1)-\phi(0)-c_1\phi'(c_1)]\\
&= [\phi'(1)-\phi'(0)] [\psi(c_1)-\psi(0)]
 \big\{ \frac{1}{2}(\psi(c_1)-\psi(0)) - c_1\psi'(c_1) \big\},
\end{align*}
and then upon utilizing the above listed properties of $\phi$ and
$\psi$ and setting $z_2=a+c_1(b-a)$,  we obtain
\eqref{flettholomorphic1}.
\end{proof}

\begin{corollary} \label{coro3.6}
Let $f$ and $h$ be holomorphic on a convex open domain $D \subset
\mathbb{C}$.  Then there exist $z_1 , z_2\in (a,b)$ such that:
(i)
\begin{equation} \label{myerholomorphic1}
\begin{split}
&[\operatorname{Im}(h'(a))] \langle b-a,
-i[h(b)-h(a)]\big\{ \frac{\langle b-a,
-i[f(b)-f(z_1)]\rangle}{\langle b-a,b-z_1\rangle }
-\operatorname{Im}(f'(z_1))\big\} \\
&= \{\operatorname{Im}[f'(b)-f'(a)]\} \langle  b-a, -i[h(z_1)-h(b)]\rangle\\
&\quad\times \big\{ \frac{1}{2}\frac{\langle b-a,
-i[h(b)-h(z_1)]\rangle}
  {\langle b-a,b-z_1\rangle } -\operatorname{Im}(h'(z_1))\big\}
\end{split}
\end{equation}
and (ii)
\begin{equation} \label{flettholomorphic}
\begin{split}
&[\operatorname{Im}(h'(b)) \langle b-a,
-i[h(b)-h(a)]\rangle] \big\{ \frac{\langle b-a,
-i[f(a)-f(z_2)]\rangle}{\langle b-a,a-z_2\rangle }
-\operatorname{Im}(f'(z_2))\big\} \\
&= \{\operatorname{Im}[f'(b)-f'(a)]\} \langle  b-a, -i[h(z_2)-h(a)]\rangle\\
&\quad\times \big\{ \frac{1}{2}\frac{\langle b-a,
-i[h(a)-h(z_2)]\rangle}
  {\langle b-a,a-z_2\rangle } -\operatorname{Im}(h'(z_2))\big\}
\end{split}
\end{equation}
\end{corollary}

\begin{proof}
 Define $f_1=-if$ and $h_1=-ih$ and note that
$\operatorname{Re}f_1'(z)= \operatorname{Im}f'(z)$ and 
$\operatorname{Re}h_1'(z)= \operatorname{Im}h'(z)$. 
Now the required   results follow at once by applying the
Theorem \ref{cauchyholomorphic} to the pair $f_1$ and $h_1$ and
rewriting them in terms of $f$ and $h$.
\end{proof}

\begin{remark} \label{rmk3.7} \rm
 By setting $h(z)=z$ in the above theorem and  corollary, and noting that
 $\frac{b-z}{b-a}= \frac{\langle b-a,b-z\rangle}{\langle b-a,b-a\rangle}$
 and $\frac{a-z}{b-a}= \frac{\langle b-a,a-z\rangle}{\langle b-a,b-a\rangle}$ 
for all $z \in (a,b)$, one gets the results
of the theorems \ref{sahootheorem2} and \ref{cttheorem2}.
\end{remark}


\begin{thebibliography}{999}

\bibitem{CT} D. Cakmak, A. Tiryaki; 
\emph{Mean Value theorem fro holomorphic
functions}, Electronic Journal of Differential Equations, Vol
2012(2012)(34), pp. 1--6.

\bibitem{DPRS} R. M. Davitt, R. C. Powers, T. Riedel,  P.K. Sahoo;
\emph{Flett's mean value
theorem for holomorphic functions},  Math. Magazine,  Vol. 72(4),  1999, pp. 304-307


\bibitem{EJ} J. C. Evard, F. Jafari;
\emph{A complex Rolle's theorem}, Amer. Math.
Monthly, Vol. 99, 1992, pp. 858-861.

\bibitem{F} T. M. Flett;
\emph{A mean value theorem}, Math. Gazette., Vol. 42, 1958, pp. 38-39.

\bibitem{MP} J. Matkowski , I.  Powlikowska;
\emph{Homogeneous means generated by mean value theorem},
 J. Mathematical inequalities, Vol.
4(4),  2010, pp. 467--479.


\bibitem{Mo} A. N.  Mohapatra;
\emph{On generalisations of some variants of Lagrange Mean Value Theorem},   
Antarctica Journal of Mathematics, Vol.  9(6), 2012, pp. 481--491.

\bibitem{mol} J. Molnarova;
\emph{On generalized Flett's mean value theorem}, arXiv.1107.3264, July, 2011,
 to appear in Int. Jour. of Math. and Math. Sc.

\bibitem{M} R. E. Myers;
\emph{Some elementary results related to mean value theorems} 
The two year college Mathematics Journal, {Vol. 8}(1),
1977,  pp. 51--53.

\bibitem{pw} I.  Powlikowska;
\emph{Stability of n-th order Flett's points and
Lagrange points}, Sarajevo J. Math, Vol. 2(14),  2008,  pp. 41--48.

\bibitem{pow2} I.  Powlikowska;
\emph{An extension of a theorem of Flett},
Demonstr. Math Vol 32, 1999, pp. 281--286.

\bibitem{RS} T. Riedel, M. Sablik;
\emph{A different version of Flett's mean
value theorem and an associated functional equation,} Acta
Mathematica Scinica, English series, Vol. 20(6), 2004,  pp.
1073--1078.

\bibitem{SD} P. K. Sahoo, D. Riedel;
\emph{ Mean value theorems and Functional Equations}, World Scientific,
 New Jersy, 1998.

\end{thebibliography}
\end{document}