Abstract
Let \(f\) be a real differentiable function in an open interval \(I\) with one-to-one derivative. We observe that if the Lagrange mean \(L^{[f]}\) of a generator \(f\) is conditionally positively homogeneous, then \(f\) must be of the class \(C^{\infty }\) and the function
is also a generator of \(L^{[f]}\) i.e. that \(L^{[g]}=L^{[f]}.\) We show that this fact and a result on equality of two Lagrange means allow easily to determine all positively homogeneous Lagrange means.
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1 Introduction
Let \(f\) be a real differentiable function defined in an open interval \( I\subset \mathbb {R}\). By the Lagrange mean-value theorem, there exists a two-place function \(L^{[f]}:I^{2}\rightarrow I\), called a Lagrange mean of a generator \(f\), such that, for all \(x,y\in I,\)
If the derivative \(f^{\prime }\) is one-to-one, then the mean \(L^{[f]}\) is uniquely determined.
It is well known that if the Lagrange mean \(L^{[f]}\) is positively homogeneous, then it is a generalized logarithmic mean \(\mathcal {L} ^{[p]}\) ([4, 5] cf. also [3], p. 404, Theorem 6). The main purpose of this note is to present a new proof of this fact. The idea is based on the observation that any generator \(f\) of a conditionally positively homogeneous Lagrange mean \(L^{[f]}:I^{2}\rightarrow I\) must be of the class \(C^{\infty }\) and that the function
is also a generator of \(L^{[f]},\) that is \(L^{[g]}=\) \(L^{[f]}.\) These facts and a result on equality of two Lagrange means (cf. [1, 6]) imply that
for some \(a,b,c\in \mathbb {R}\). Solving this differential equation we obtain all the homogeneous Lagrange means.
In section 2 we recall the definitions of a mean, the Lagrange mean based on Lagrange’s mean-value theorem (that is used in this paper), its integral counterpart and some of its properties. In section 3 we prove Theorem 3.2, a slight modification of the results on the equality of two Lagrange means (cf. [1] and [6]). In the first part of Theorem 3.2 we assume that at least one of two equal means is uniquely determined. In section 4 we show that any generator \(f\) of the conditionally homogeneous Lagrange mean \(L^{[f]}\) in an open interval \(I,\) that is such that, for all \(x,y\in I\) and \(t>0\),
is infinitely times differentiable (Theorem 4.1). Applying these results, in section 5 we determine all the conditionally homogeneous Lagrange means (Theorem 5.1).
Applicability of the method to determine the form of all conditionally homogeneous Cauchy means is discussed in the last section.
2 Means, Lagrange means and some their properties
Let \(I\subset \mathbb {R}\) be an interval. A function \(M:I^{2}\rightarrow \mathbb {R}\) is said to be a mean in \(I\) if
A mean \(M\) is called strict if these inequalities are sharp for all \(x,y\in I\), \(x\ne y;\) and symmetric if \(M\left( x,y\right) =M\left( y,x\right) \) for all \(x,y\in I.\)
Remark 2.1
If \(M:I^{2}\rightarrow \mathbb {R}\) is a mean in an interval \(I,\) then
-
(1)
\(M\) is reflexive, that is
$$\begin{aligned} M\left( x,x\right) =x, \quad \quad x\in I; \end{aligned}$$ -
(2)
for any interval \(J\subset I,\)
$$\begin{aligned} M\left( J^{2}\right) =J; \end{aligned}$$in particular, \(M\left( I^{2}\right) =I.\)
Remark 2.2
If \(M:I^{2}\rightarrow \mathbb {R}\) is a reflexive function, that is if \( M\left( x,x\right) =x\) for all \(x\in I,\) and (strictly) increasing with respect to each variable, then it is a (strict) mean in \(I.\)
Remark 2.3
If \(M:I^{2}\rightarrow I\) is a continuous and strict mean in \(I,\) then for any \(u\in \hbox {int}\;I\) there are \(x,y\in \hbox {int}\;I,\) \(x\ne y,\) such that \(M\left( x,y\right) =u.\)
By the Lagrange mean-value theorem, if a real function \(f\) defined on an interval \(I\subset \mathbb {R}\) is differentiable, then there exists a mean \( L^{[f]}:I^{2}\rightarrow I\) such that, for all \(x,y\in I,\) \(x\ne y,\)
If moreover \(f^{^{\prime }}\)is one-to-one, then
so \(L^{[f]}\) is uniquely determined and, by the Darboux property of derivative, \(f^{\prime }\) is continuous and strictly monotonic (cf. [7], Remark 2.1), so \(f\) is either strictly convex or strictly concave in \(I\).
The mean \(L^{[f]}\) is called the Lagrange mean and the function \(f\) its generator. It obvious that \(L^{[f]}\) is continuous, symmetric and strict, that is
Remark 2.4
Assume that \(h:I\rightarrow \mathbb {R}\) is continuous and strictly monotonic. Then \(\mathcal {L}^{h}:I^{2}\rightarrow I\) defined by
is a strict mean in the interval \(I\) and it is called a Lagrange mean (cf. for instance [1, 2], [3]). Setting \( h:=f^{\prime },\) it is not difficult to see that \(\mathcal {L}^{h}=L^{[f]}.\) Note that in the integral definition of the Lagrange mean \(\mathcal {L}^{h}\) from the very beginning it is assumed that the function \(h\) (the derivative of \(f\)) is continuous and strictly monotonic.
3 Equality of two Lagrange means
We need the following [cf. [6, Theorem 3.2]
Lemma 3.1
Let \(I\subset \mathbb {R}\) be an open interval, \(f,g:I\rightarrow \mathbb {R}\) , and let \(I_{f}\) stand for the range of the two variable function
Suppose that \(f\) is strictly convex or strictly concave on \(I\). If \(\phi :I_{f}\rightarrow \mathbb {R}\) satisfies the functional equation
then there exist \(a,b\in \mathbb {R}\) such that
Applying this lemma we prove the following (cf. [6])
Theorem 3.2
Let \(I\subset \mathbb {R}\) be an interval. Suppose that the functions \( f,g:I\rightarrow \mathbb {R}\) are differentiable.
(i) If \(f^{\prime }\) is one-to-one and
then there are \(a,b,c\in \mathbb {R}\), such that
(ii) If \(f^{\prime }\) and \(g^{\prime }\) are one-to-one and
then there are \(a,b,c\in \mathbb {R}\), \(a\ne 0,\) such that
Proof
To prove (i) assume that \(f^{\prime }\) is one-to-one and \( L^{[g]}=L^{[f]}\). Since
and
we obtain
whence, setting
we get the equality
By the Darboux property of derivative, \(f^{\prime }\) is continuous and strictly monotonic (cf. [7], Remark 2.1), so \(f\) is either strictly convex or strictly concave in \(I\). In view of Lemma 3.1 there exist \( a,b\in \mathbb {R}\) such that
whence, by (3.1),
which implies that
Consequently, there is \(c\in \mathbb {R}\) such that
which completes the proof of (i). If \(g^{\prime }\) is one-to-one then, clearly, in this formula the number \(a\) cannot be zero.\(\square \)
Remark 3.3
Assuming the integral definition of the Lagrange mean, Berrone & Moro proved [1, Corollary 7] that if \(f^{\prime }\) and \(g^{\prime }\) be continuous and strictly monotonic functions defined on an interval \(I\), then their corresponding Lagrange means coincide if, and only if, there are two real constants \(\gamma _{1},\gamma _{2},_{\text { }}\)with \(\gamma _{1}\ne 0\) such that
In fact, part (ii) of Theorem 3.2 is equivalent to the result of Berrone & Moro.
4 The homogeneity of \(L^{[f]}\) implies the high regularity of \(f\)
In this section we prove the following
Theorem 4.1
Let \(I\subset \left( 0,\infty \right) \) be an open interval. Suppose that \( f:I\rightarrow \mathbb {R}\) is differentiable and \(f^{\prime }\) is one-to-one. If the Lagrange mean \(L^{[f]}\) is conditionally positively homogeneous, that is, if
for all \(x,y\in I\) and \(t>0\) such that \(tx,ty\in I,\) then \(f\) is of the class \(C^{\infty }\) in \(I.\)
Proof
Assume that \(L^{[f]}\) is conditionally positively homogeneous. Then, by the definition of the Lagrange mean,
for all \(x,y\in I,\) \(x\ne y,\) and \(t>0\) such that \(tx,ty\in I.\) Take arbitrary \(u\in I.\) According to Remark 2.3 there exist \(x,y\in I,\) \(x\ne y,\) such that \(L^{[f]}\left( x,y\right) =u.\) From (3.1) we get
Since the right-hand side is a differentiable function of \(t\) in a neighborhood of \(1,\) the function \(f^{\prime }\) is differentiable in a neighborhood at the point \(u.\) Since \(u\in I\) is arbitrary, the function \(f\) is twice differentiable in \(I.\) Now an easy inductive argument shows that \(f\) is of the class \(C^{\infty }\) in \(I\).\(\square \)
5 Equality of means as an approach to determine the homogeneous Lagrange means
We apply Theorem 3.2 on the equality of two Lagrange means to prove the following
Theorem 5.1
Let \(I\subset \left( 0,\infty \right) \) be an open interval. Suppose that \( f:I\rightarrow \mathbb {R}\) is differentiable and \(f^{\prime }\) is one-to-one. Then the following conditions are equivalent:
-
(i) the Lagrange mean \(L^{[f]}\) is conditionally positively homogeneous, that is,
$$\begin{aligned} L^{[f]}\left( tx,ty\right) =tL^{[f]}\left( x,y\right) \end{aligned}$$(5.1)for all \(x,y\in I\) and \(t>0\) such that \(tx,ty\in I;\)
-
(ii) there are \(p,a,b,c\in \mathbb {R}\), \(p\ne 1,\) \(a\ne 0\), such that
$$\begin{aligned} f(x)=\left\{ \begin{array}{lll} ax^{p}+bx+c, &{}\quad x\in I &{}\quad if \; 0\ne p\ne 1 \\ a\ln x+bx+c, &{}\quad x\in I &{}\quad if \; p=0 \\ ax\ln x+bx+c &{}\quad x\in I &{}\quad if \; p=1 \end{array} \right. ; \end{aligned}$$ -
(iii) there is \(p\in \mathbb {R}\) such that \(L^{[f]}=\mathcal {L} ^{[p]}|_{I^{2}}\) where \(\mathcal {L}^{[p]}:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) \) is defined by
$$\begin{aligned} \mathcal {L}^{[p]}\left( x,y\right) :=\left\{ \begin{array}{ll} \left( \frac{x^{p}-y^{p}}{p\left( x-y\right) }\right) ^{1/\left( p-1\right) }&{}\quad if \; p\ne 0,\; p\ne 1 \\ \frac{x-y}{\ln x-\ln y} &{}\quad if \; p=0 \\ e^{-1}\left( \frac{x^{x}}{y^{y}}\right) ^{1/\left( x-y\right) } &{}\quad if \; p=1 \end{array} \right. for \; x\ne y;\quad \mathcal {L}^{[p]}\left( x,x\right) =x. \end{aligned}$$(5.2)
Proof
To show the implication \((i)\Rightarrow (ii)\) assume \((i)\). By Theorem 4.1 the function \(f\) is twice differentiable in \(I.\) Since \(f^{\prime }\) is one-to-one, the set \(\left\{ x\in I:f^{\prime \prime }\left( x\right) =0\right\} \) is nowhere dense in \(\left( 0,\infty \right) \). Let \(J\subset I\) be a maximal (in the sense of inclusion) open interval such that \(f^{\prime \prime }\left( x\right) \ne 0\) for all \(x\in J\). Since the interval \(J\) is open, for arbitrarily fixed \(x,y\in J,\) \(x\ne y,\) there is \(\delta >0\) such that for all \(t\in \left( 1-\delta ,1+\delta \right) \) we have \(tx,ty\in J,\) whence \(L^{[f]}\left( tx,ty\right) \in J\) and, consequently,
Differentiating both sides of (5.1) with respect to \(t\in \left( 1-\delta ,1+\delta \right) ,\) we get
that is
whence, setting \(t=1,\) after obvious simplification, we obtain
Since the function \(g:J\rightarrow \mathbb {R}\) defined by
is differentiable and
we can write this equality in the form
which implies that \(L^{[g]}=L^{[f]}\) in \(J^{2}\). By Theorem 3.2(i) there are \( \alpha ,\beta ,\gamma \in \mathbb {R}\), such that
In view of (5.3) we hence get
whence, setting
we obtain
Solving this linear differential equation, in the case when \(0\ne p\ne 1,\) we get
in the case when \(p=0,\) we get
and in the case when \(p=1,\) we get
for some \(a,b,c\in \mathbb {R}\). Since, by the assumption, \(f^{\prime }\) is one-to-one, we conclude that the number \(a\) cannot be zero. Note also that in the first case
in the second case when \(p=0\),
and, in the case \(p=1,\)
Since \(J\) is a maximal subinterval of \(I\) such that \(f^{\prime \prime }\left( x\right) \ne 0\) for all \(x\in J,\) it follows that \(J=I\) for all \(p.\) This proves that \((ii)\) holds true.
Since the remaining implications \((ii)\Rightarrow (iii)\) and \( (iii)\Rightarrow (i)\) are easy to verify, the proof is complete.\(\square \)
Remark 5.2
Clearly, for any \(p\in \mathbb {R}\), the mean \(\mathcal {L}^{[p]}\) is posiyively homogeneous, that is
The mean \(\mathcal {L}^{[0]}\) is called the logarithmic mean, the mean \( \mathcal {L}^{[1]},\) defined as the pointwise limit, by the formula
is called the identric mean, and \(\left\{ \mathcal {L}^{[p]}:p\in \mathbb {R} \right\} \) is sometime called the family of generalized logarithmic means. It is well-know (and obvious) that, for arbitrary \(\left( x,y\right) \in \left( 0,\infty \right) ^{2},\) the function
is continuous.
6 Remarks on the relevant question for Cauchy means
Let \(I\subset \mathbb {R}\) be an interval. Assume that \(f,g:I\rightarrow \mathbb {R}\) are differentiable and \(g^{\prime }\left( x\right) \ne 0\) for all \(x\in I.\) By the Cauchy mean-value theorem, there exists a Cauchy mean \(C^{[f,g]}:I^{2}\rightarrow I\) such that, for all \(x,y\in I,\) \( x\ne y,\)
(The functions \(f\) and \(g\) are called the generators of \(C^{[f,g]}.\)) If moreover \(\frac{f^{\prime }}{g^{\prime }}\) is one-to-one, then
so \(C^{[f,g]}\) is uniquely determined. Note here that
In [7], Proposition 1, it was observed that
that is the Cauchy mean \(C^{[f,g]}\) is \(g\)-conjugate to the Lagrange mean \( L^{[f\circ g^{-1}]}\) of the generator \(f\circ g^{-1}.\) Hence we have
Now applying Theorem 5.1 with \(f\) replaced by \(f\circ g^{-1}\) we obtain the following
Corollary 6.1
Let \(I\subset \left( 0,\infty \right) \) be an open interval. Suppose that \( f,g:I\rightarrow \mathbb {R}\) are differentiable, \(g^{\prime }\left( x\right) \ne 0\) for all \(x\in I,\) and \(\left( \frac{f^{\prime }}{g^{\prime }}\right) ^{-1}\) is one-to-one. Then the following conditions are equivalent:
(i) the Lagrange mean \(L^{[f\circ g^{-1}]}\) is conditionally positively homogeneous, that is,
for all \(u,v\in I\) and \(t>0\) such that \(tu,tv\in g\left( I\right) ;\)
(ii) there are \(p,a,b,c\in \mathbb {R}\), \(p\ne 1,\) \(a\ne 0\), such that
(iii) there is \(p\in \mathbb {R}\), \(p\ne 1,\) such that
where \(\mathcal {L}^{[p]}:\left( 0,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) ,\) for every \(p\in \mathbb {R}\), is defined by (5.2).
In this context it is natural to ask if we can determine all conditionally positively homogeneous Cauchy means by a similar method as that in the case of Lagrange means. It turns out that, in general not. First of all let us note that, under the assumptions of the definition, the condition
for all \(x,y\in I\) and \(t>0\) such that \(tx,ty\in I,\) does not imply that both \(f\) and \(g\) must be of the higher regularity. It is not difficult to see that this condition and the assumption that one of the functions of \(f\) or \(g\) is twice differentiable imply that the remaining one is also twice differentiable.
Assuming that one of these functions is twice differentiable and differentiating both sides of (6.1) with respect to \(t\) and then setting \(t=1\) leads to the following interesting equality
for all \(x,y\in I,\) \(x\ne y.\) Hence, setting
for a differentiable function \(h\) such that \(h^{\prime }\left( u\right) \ne 0\) for all \(u\in I,\) and applying the Cauchy mean-value theorem for each of two difference quotient on the right side, we obtain
Thus we have shown the following
Corollary 6.2
Let \(I\subset \left( 0,\infty \right) \) be an open interval. Suppose that \( f,g:I\rightarrow \mathbb {R}\) are differentiable, \(g^{\prime }\left( x\right) \ne 0\) for all \(x\in I,\) and \(\left( \frac{f^{\prime }}{g^{\prime }}\right) ^{-1}\) is one-to-one. If the Cauchy mean \(C^{[f,g]}\) is conditionally positively homogeneous, then (6.2) holds true.
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Matkowski, J. On homogeneous Lagrange means. Period Math Hung 68, 119–127 (2014). https://doi.org/10.1007/s10998-014-0015-6
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DOI: https://doi.org/10.1007/s10998-014-0015-6