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\centerline{ MEANS GENERATING THE CONE SECTIONS }

\centerline{  AND THE THIRD DEGREE POLYNOMIALS}

\vskip6pt

\centerline{\it Kalnitsky V. S.}
\centerline{\it skalnitsky@hotmail.com}

\vskip12pt

\centerline{\bf Introduction}

\vskip6pt

Historically the first "rich" class of means appeared from the
Lagrange mean value theorem. This class consists of so called
$f$-differential means $$
D_f(u,v)=\left({f'}^{-1}\right)\left(\frac{f(u)-f(v)}{u-v}\right)
$$ and after Stolarsky's generalization of logarithmic mean
$D_{t^\alpha}$ [1], named after his name, was intensive
investigated.

After that new class of quasi-arithmetic means was introduced
$$
M_f(u,v)=f^{-1}\left(\frac{f(u)+f(v)}{2}\right).
$$

And as the first comparison theorem of classes we can count
G.~Aumann's theorem which states that all solutions of the
differential-functional equation
$$
D_f=M_F
$$
are arcs of cone sections. This result was generalized by
Sh.~Haruki (see, for inst., [2]). It gave the impact to the
investigation of functional equations of type
$$
\frac{f(x)-f(y)}{x-y}=\varphi(\xi(x,y)),
$$
so called Stemet type equations [3].

Recently many rich classes of means are described: Tobey's
homoigeneous means~[4] and their generalizations~[5], Toader's
means~[6], $\alpha,\beta$-means~[7], functional Stolarsky means~[8].

In the works of J.~S\'andor~[9] and M.~Kuczma~[10] was noted
that all monotone convex (vertex) solutions of equation
$$
\frac{f(x)-f(y)}{x-y}=\varphi(h(x),h(y)),
$$
where $\varphi(u,v)=u+v;uv;1/(u+v)$, are arcs of cone sections.
Assuming $h,\varphi\in C$, $\varphi(u,u)$ being inversable, the
above equation is reducable to the form
$$
\frac{f(x)-f(y)}{x-y}=K(f'(x),f'(y)),\eqno(1)
$$
where $K$ is necessarily generalized mean. For $K=\frac{u+v}2$
it is the Lagrange theorem of $2^{nd}$ degree
polinomials characterisation.

In this form Kuczma's result can be formulated in the following
way: all solutions of equation
$$
\frac{f(x)-f(y)}{x-y}=\frac{f'(x)+f'(y)}2;\quad\sqrt{f'(x)f'(y)};
\quad\frac{2f'(x)f'(y)}{f'(x)+f'(y)}
$$
are vertical paraboles, arcs of increasing vertical hyperboles
and horizontal paraboles correspondently.

The aim of present article is to describe all means $K(u,v)$
generating as solutions arcs of the cone sections.

The equation (1) itself is found out have relations with different
disciplines such as economics~[11], operational research~[12,13]
and analysis~[14], low-dimesional geometry~[15].
\vskip12pt

\centerline{\bf \S1 Classification theorem}

\vskip6pt

We consider the class of functions $f(t)\in C^1$, convex
(vertex) on its domain of definition, $K(u,v)$\,--- smooth
symmetrical generalized mean. Let $f(t)$ be a solution of
equation~(1), the transformations of function
$$
f(t+\alpha)+\beta;\quad\frac1kf(kt),
$$
where $\alpha,\beta,k\in I\!\!R$, are solutions as well. So we
can assume without lose of generality that
$$
f(t):[0,1]\rightarrow I\!\!R,\quad f(0)=0,
$$

{\bf Lemma}. {\it If for given $K(u,v)$ and $C$ the function
$\theta(u)=K(u,C)\in C^1$, then the monotone convex (vertex)
solution $f(t)$ of the equation~(1)
if exists
is unique up to the homotety. }

P r o o f. As the function $K(u,v)$ is symmetrical mean $K(C,C)=C$
and $K_u(C,C)=1/2$, i.e. for some interval the function
$\theta(u)$ is monotone (therefore inversable)
and the only solution of equation
$\theta(u)=u$ is $C$.
For given initial data we can write
$f'(x)=s(f(x)/x)$, where $s(u)=\theta^{-1}(u)\in C^1$.
The statement of the lemma
follows from the fact: for  any $s(t)$ such that $s(c)=c$ is the
only solution and $s'(C)=2$ the solution of the above homogeneous
ODE  exists, belongs to $C^2$, has the form
$$
f(t)=\frac{(\alpha t)F^{-1}(\ln(\alpha t))}{\alpha},
$$
where $\alpha\in I\!\!R$,
$F(z)=\int^a_z\frac{d\mu}{s(\mu)-\mu}$,
and is determined up to the $f''(0)$.
Geometrically it means the uniqueness up to the homotety. Note,
$f'(0)=C$. Let's
now consider the found solution $f(t)$ and construct the mean
$K_f$ by
the formula
$$
{K_f}(u,v)=\frac{f({f'}^{-1}(u))-f({f'}^{-1}(v))}{{f'}^{-1}(u)-{f'}^{-1}(u)}
$$
If $K_f=K$ the function $f(t)$ is the solution of the
equation (1).~$\Box$

Let's now derive some relations between $K_f$ and $K_g$ for $f$
and $g$ related somehow.

Let $f(t)$ be a solution of (1) and $g(t)=f(t)+kt$. We can write
$$
K_g(g'(t),g'(s))=\frac{g(t)-g(s)}{t-s}=\frac{f(t)-f(s)}{t-s}+k=
$$
$$
=K_f(f'(t),f'(s))+k=K_f(g'(t)-k,g'(s)-k)+k.
$$
Therefore
$$
K_g(u,v)=K_f(u-k,v-k)+k.
$$
Geometrically, described transformation is the affine map of plane
$(x,y)\mapsto(x,y+kx)$.

Let's consider now $(x,y)\mapsto(x,ky)$, i.e. $g(t)=kf(t)$.
$$
K_g(g'(t),g'(s))=kK_f(f'(t),f'(s))=k_f(g'(t)/k,g'(s)/k),
$$
i.e.
$$
K_g(u,v)=kK_f\left(\frac uk,\frac vk \right).
$$
Analogously, for $g(t)=f(t/k)$ we have
$$
K_g(u,v)=\frac1kK_f(ku,kv).
$$

For transformation $(x,y)\mapsto(y,x)$, i.e. $g(t)=f^{-1}(t)$,
we can easy derive the
relation
$$
K_g(u,v)=\frac1{K_f\left(\frac1u,\frac1v\right)}.
$$

The compositions of all described maps generate all affine
transformations of plane. For such one with the
matrix
$$
A=\left(
\begin{array}{cc}
\alpha&\beta\\
\gamma&\mu\\
\end{array}\right);\quad \alpha\mu-\beta\gamma\ne 0.
$$
we consider the function  $r(t)=\frac{\mu t+\gamma}{\beta
t+\alpha}$. For any $f(t)$ and its $A$-image $g(t)$
$$
K_g(u,v)=r(K_f(r^{-1}(u),r^{-1}(v))),
$$
so called $r$-conjugancy of $K$ ($r^*K$).

Let's denote
$$
K_a=\frac{u+v}2;\quad K_g=\sqrt{uv};\quad K_c=
\frac{\sqrt{(u^2+1)(v^2+1)}+uv-1}{u+v}
$$
and
$$
\Lambda=\left\{r^*K|K=K_a,K_g,K_c\right\}.
$$

{\bf Theorem.} {\it For given $K\in\Lambda$ all monotone convex
(vertex) solutions of the equation (1) are arcs of paraboles,
hyperboles or ellipses.}

P r o o f. Let's consider $K_a,K_g,K_c$ at first.

From the Lagrange theorem it follows that all solutions of equation
(1) for $K_a$ are vertical paraboles.

From the Kuczma's theorem it follows that all solutions of equation
(1) for $K_g$ are arcs of increasing vertical hyperboles.

Let's now consider the arc of cycle $f(t)=\sqrt{1-t^2},\ t\ge0$.
Streightforward calculation gives $K_f=K_c,\ u,v\le0$. For three others
arcs of cycle the means are the same, as $-K_c(-u,-v)=K_c(u,v)$
for $uv\ge0$. Any ellipse on plane can be devided on four
arcs of monotonisity. There is an affine transformation mapping
these arcs to the arcs of monotonisity of canonical cycle. Therefore
the arcs of ellipse generates the means conjugated to
$K_c$.
Note, for points of cycle belonging to the different arcs of it
the relation is not $K_g$ and no mean at all.

To finish the proof note that affine images of described curves
are arbitrary cone sections and means generated by them are from
$\Lambda$.~$\Box$

\vskip12pt

\centerline{\bf \S2. 3${}^d$ degree polynomials
characterisation}


\vskip12pt

It is naturally to ask about the functional-differential
equation characterising the polynomials of degree three. This
question is answered for the arbitrary degree by G.~Gross~[16].
But the form of it is not symmetrical. We will describe
one-parameter family of means generating the 3d degree
polynomials as solution of equation~(1).

By Lemma it is enough to describe $K_p$, where $p$ is
polynomial up to shifts and homotety. Let's consider the
polynomial
$$
p(t)=a_3t^3+a_2t^2+a_1t+a_0.
$$
Substitute $t=s-a_2/(3a_1)$ and adding some constant reduce it
to the form
$$
p(s)=a_3s^3+cs,
$$
where $c=a_1-a_2^2/(3a_3)$. For $c=0$
$$
K_p(u,v)=H(u,v)=\frac{u+\sqrt{uv}+v}{3}
$$
Heron's mean. For arbitrary $c$
$$
K_p(u,v)=H(u-c,v-c)+c=\frac{\sqrt{(u-c)(v-c)}+u+v+c}3.
$$
Therefore we proved the following result

{\bf Theorem.} {\it All monotone convex (vertex) solutions of
equation
$$
\frac{f(t)-f(s)}{t-s}=\frac{\sqrt{(f'(t)-c)(f'(s)-c)}+f'(t)+f'(s)+c}3
$$
are arcs of polynomials
$$
p(t)=a_3t^3+a_2t^2+a_1t+a_0,
$$
for which $c=a_1-a_2^2/(3a_3)$.
}

{\bf References}

[1] Stolarsky K.B. Generalization of the logarithmic mean
//~Math.~Mag. {\bf 48}~(1975); pp.~87--92.

[2] Acz\'el J. A mean value property of the derivative of
quadric polynomials without mean values and derivatives
//~Math.~Mag. {\bf 58}~(1985),  N~1, pp.~42--45.

[3] Acz\'el, J., Kuczma, M. On two mean value properties and
functional equations associated with them //~Aequat. Math. {\bf
38}(1989),  N~2--3.

[4] Tobey, M.O. Two-parameters homogeneous mean value //~Proc.
Amer. Math. Soc. {\bf 18}(1967).

[5] Pe\v caric, \v Simi\'c. Stolarsky-Tobey mean in n-variables
//~Math. Ineq. Appl.  N~2(1999).

[6] Toader, Gh. Some mean values related to the
arithmetic-geometric mean //~J. Math. Anal. and Appl. 2/8,
 N~2(1998), pp.~358--368.

[7] Grossman, M., Katz, R. A new approach to means of two positive
numbers //~Int. J. Math. Educ. Sci. and Techn. {\bf 17}(1986),
N~`2, pp.~205--208.

[8] Pearce, Pe\v caric, \v Simi\'c. Functional Stolarsky means
//~Math. Ineq. and Appl. V.~2(1999),  N.~4.

[9] S\'andor, I. On certain functional equations /~Itinerat
Seminar on Func. Eq. Approx. and Convexity. (Cluj-Napoca, 88).
Preprint, 88--6, Uni. "Babes-B."

[10] Kuczma, M. On some functional equations with conic sections
as solutions //~Roczniki Polskiego Towar. Mat., Seria~5, Dydaktyka
Mat.  N~13(1993).

[11] Akerman, E., Kalnitsky, V. The purpose function in the
problem of the effective distribution of resurces //Works of
Int. Sci. School "MA SRQ 2001", 18--22 June 2001, St.P., Russia.

[12] Kalnitsky, V. The reconstruction operator of two-argument
function by its section /~Kazan Math. Soc. "Lobachevsky
readings--2001" //~Proc. of Int. Sci. Conf. (28/11--01/12/2001),
Kazan.

[13] Kalnitsky, V. Unsolvableness of the Lagrange equation for the
H\"older mean //~Vestn. of St. Petersburg State University, Ser.
1(2002),  N~1, pp.~13-14.


[14] Kalnitsky, V. Generalized Lagrange differential-functional
equation. //~Proc. of Int. Conf. 10th Summer St. P. Meetings in
Analysis. August 22--26, 2001. Russia

[15] Kalnitsky, V. Arithmetic properties of the Lagrange
equation solutions. //Proc. of ICM'2002, August 20--28, Beijing,
China.

[16] Gros, G.E. A functional identity characterizing polinomials.
//~Aequat. Math. {\bf 34}(1987),  N~2--3, pp.~147--152.





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