Solving Invariant Problem of Cauchy Means Based on Wronskian Determinant ()
1. Introduction
Throughout this paper, let
be a nonempty open interval. In the sequel, the classes of continuous strictly monotone and continuous positive real-valued functions defined on I will be denoted by
and
, respectively.
Let
be an open interval. A two-variable function
is called a mean on the interval I if
holds. If for all
,
, these inequalities are strict, M is called strict ([1]). Obviously, if M is a mean, then M is reflexive, i.e.,
for all
.
Let
be means. A mean
is called invariant with respect to the mean-type mappings
, shortly,
-invariant ([2]), if
The simplest example of when the invariance equation holds is the well-known identity
where
are arithmetic mean, geometric mean and harmonic mean respectively.
The Cauchy mean
is defined by
are two continuous functions such that
and
is bijective.
In 1984, Leach and Sholander [3] proposed the Cauchy mean theorem for difference quotients and the definition of Cauchy mean. Losonczi solved the comparison and equality problems of Cauchy means for more than two variables and the equality problem for the two-variable Cauchy means under the assumption of seven times differentiability in [4]-[6]. Then Matkowski reduced the regularity condition for the equality problem for the two-variable Cauchy means to first-order differentiability [7]. Other definitions for Cauchy have been in [8] [9]. The invariant equations of some special Cauchy mean-type mappings with geometric mean or arithmetic mean have been studied in [10] [11]. Other invariance of more generalized means generated by measure integral have been considered in [12] [13]. Recently, Zhang [14] gave a survey of results dealing with the problem of means.
In this paper, we will study the invariant equation of Cauchy mean with respect to the arithmetic mean, that is to solve
(1.1)
where
,
are continuous functions such that
and
are injective functions.
2. Auxiliary Results
In order to describe the regularity conditions related to the two unknown functions
generating the mean
, we introduce some notations. The class
consists of all those pairs
of continuous functions
such that
and
. For
, we say that the pair
is in the class
if
are n-times continuously differentiable functions such that
and the function
does not vanish anywhere on I. Obviously, the latter condition implies that
is strictly monotone, i.e.,
.
For
, we also introduce the notation
where the
-order Wronskian operator
is defined in terms of ith and jth derivatives by
Lemma 1 (Lemma 5 in [15]) Let
. Then
are solutions of the second-order differential equation
In what follows, we will give some formulae for the high-order directional derivatives of
at the diagonal points of the Cartesian product
. Given a pair
and a fixed element
, define the function
in a neighborhood of origin by
(2.1)
Using Lemma 2.2 from [16], we get the following
Lemma 2 Let
,
. Then, for fixed
, the function
defined by (2.1) is n-times continuously differentiable at the origin and
Furthermore,
for
and for the cases
, we have
Lemma 3 If
, and Equation (1.1) holds, then
(2.2)
Proof. By Equation (2.1), the invariant Equation (1.1) can be rewritten by
(2.3)
By differentiating the above equation twice with respect to variable u, it can be obtained that
letting
And because
then
that is
Lemma 4 If
, and Equation (1.1) holds, then
(2.4)
Proof. By differentiating Equation (2.3) fourth with respect to variable u, it can be obtained that
letting
, we have
Since
then we get
3. Solutions to Invariant Equations under Special Conditions
Although it is difficult to directly solve the Cauchy mean invariant equation with respect to the arithmetic mean, some types of invariant equations can also be solved with certain prerequisites. Next, we will solve the Cauchy mean invariant equation with respect to the arithmetic mean when the denominator functions
are equal and satisfy the harmonic oscillator equation
(3.1)
for some
. Then, we introduce the sine and cosine type functions
by
Due to basic results on the second-order linear differential equations, the functions
and
given above form a fundamental system of solutions for the differential Equation (3.1).
Theorem 1 If
. Assume
satisfying Equation (3.1) and Equation (1.1) holds, then exists
such that
(3.2)
Proof. Since Equation (3.1) holds, by the definitions of
and
we get
Similarly, we have
Using (2.2) and the above two equalities, we have
and thus
. Consequently, (2.4) becomes
which implies that there exists some
such that
Hence, we obtain
Since
and
, integrating the above equations, we get (3.2).
In what follows, we will restrict to the basic solutions to the equation of harmonic oscillator, i.e. consider the functions
(H1)
for
;
(H2)
for
;
(H3)
for
.
Theorem 2 If
. Assume
satisfying one of the conditions (H1) - (H3) and Equation (1.1) holds, then
1) For case (H1), there exists
,
2) For case (H2), there exists
,
3) For case (H3), there exists
,
Next, we will consider the solution of the invariant equation for (1.1) when the denominator functions are power functions, that is
,
.
Theorem 3 Let
and the invariant Equation (1.1) holds, then there exists
such that
where
.
Proof. When
we have
and
, the relationship between
and
is
Similarly
Using (2.2) and the above two equalities, we have
and thus
Consequently, (2.4) becomes
which implies that there exist
such that
Hence, we obtain
By
and
, integrating the above equations, we get the result.