Quasi-Mean Value Theorem

• Quasi-Mean Value Theorem is a generalization of the classical MVT where the mean-point is fixed as a quasi-arithmetic mean defined via a strictly monotone function.
• It classifies pairs of differentiable functions into four families—linear, quadratic, exponential, and trigonometric—based on their functional forms.
• The approach transforms the problem into a functional-differential equation, offering insights into mean-type function equations and applications in analysis.

The Quasi-Mean Value Theorem (QMVT) encompasses a family of results generalizing the classical Cauchy Mean Value Theorem (MVT) by prescribing a specific, often nonlinear, mean as the location where the mean value identity holds. This extension is realized through quasi-arithmetic means, allowing the theorem to characterize all pairs of differentiable functions for which the mean-value identity is rigidly achieved at a predetermined mean of two arguments. The resulting functional-differential equations lead to an explicit classification of solution families, making the QMVT a genuinely robust generalization of MVT theory, with connections to mean-type function equations and Taylor polynomial intersection means (Ibragimov, 2020, Balogh et al., 2015, Horwitz, 2014).

1. Formal Statement and Generalization

Let $E\subset\mathbb{R}$ be a nonempty open interval and $H:E\to H(E)$ a differentiable, strictly monotone function with inverse $H^{-1}$. Fix $\alpha,\beta\in(0,1)$ with $\alpha+\beta=1$. Define the quasi-arithmetic mean $h(x,y)=H^{-1}(\alpha H(x)+\beta H(y))$ for $x,y\in E$. The Quasi-Mean Value Theorem seeks all pairs of differentiable functions $\varphi,\psi:E\to\mathbb{R}$ satisfying

$$[\varphi(y) - \varphi(x)]\,\psi'\bigl(h(x,y)\bigr) = [\psi(y) - \psi(x)]\,\varphi'\bigl(h(x,y)\bigr) \tag{QMVT-Eq}$$

for all $x<y$ in $E$. Unlike the classical Cauchy MVT, which guarantees the existence of an unspecified $c\in(x,y)$, the QMVT fixes $c$ to the prescribed mean $h(x,y)$ (Ibragimov, 2020).

2. Solution Classification

The solution space to (QMVT-Eq) is completely described by four mutually exclusive families:

Type	Functional Form	Additional Details
Linear dependence	$A+B\,\varphi(x)+C\,\psi(x)\equiv 0$ for some $A,B,C$, not all zero	Functions are linearly dependent
Quadratic-type	$$\varphi(x),\psi(x)\in \operatorname{span}\{1, H(x), H(x)^2\}$$	$H$ arbitrary monotone, e.g., identity or logarithm
Exponential-type	$$\varphi(x),\psi(x)\in \operatorname{span}\{1, e^{\mu H(x)}, e^{-\mu H(x)}\},\, \mu\ne 0$$	Admits real exponential growth/decay terms
Trigonometric-type	$$\varphi(x),\psi(x)\in \operatorname{span}\{1, \sin(\mu H(x)), \cos(\mu H(x))\},\, \mu\ne 0$$	Oscillatory structure; $\mu$ real

This systematic classification generalizes the classical polynomial, exponential, and trigonometric cases observed in the symmetric Cauchy MVT, extending them to the nonlinear regime via $H$ (Ibragimov, 2020, Balogh et al., 2015). When $H(t)=t$, the mean specializes to the weighted arithmetic mean; for $H(t)=\ln t$ on $(0,\infty)$, to the geometric mean, yielding quadratic, logarithmic, exponential, and trigonometric families as solutions.

3. Proof Methodology

The proof exploits the strict monotonicity of $H$, translating the original equation via the substitution $u=H(x), v=H(y)$ into

$$[F(v) - F(u)] G'(\alpha u + \beta v) = [G(v) - G(u)] F'(\alpha u + \beta v)$$

where $F(u) = \varphi(H^{-1}(u))$, $G(u) = \psi(H^{-1}(u))$. The problem reduces to a functional-differential equation where the mean-point is the linear mean in $u,v$. The argument splits into "asymmetric" ($\alpha\neq\frac12$) and "symmetric" cases:

• For $\alpha\neq\frac12$, only linear dependence is possible (Balogh et al., 2015).
• For $\alpha=\frac12$, a bootstrapping argument induces $C^\infty$ regularity and reduces the problem to classifying $C^\infty$ solutions of

$[F(x+h)-F(x-h)]G'(x) = [G(x+h)-G(x-h)]F'(x)$

which must fall into one of the three functional families (quadratic, exponential, trigonometric). These transfer back to the original variable via $H^{-1}$ (Ibragimov, 2020, Balogh et al., 2015).

4. Notable Instances and Examples

• Arithmetic Mean: $H(t)=t$, $h(x,y)=\alpha x+\beta y$. Recovers earlier results by Balogh–Ibragimov–Mityagin and Łukasik (Balogh et al., 2015).
• Geometric Mean: $H(t)=\ln t$ on $E=(0,\infty)$, $h(x, y) = x^\alpha y^\beta$. The solutions become linear combinations of $1$, $\ln x$, $(\ln x)^2$, exponentials in $\ln x$ (i.e., powers of $x$), and trigonometric functions of $\ln x$.
• Taylor Polynomial Means: For $f\in C^{r+1}$ with $r$ odd and $f^{(r+1)}$ of fixed sign, there is a unique $x_0\in(a,b)$ such that $P_a(x_0) = P_b(x_0)$ for the $r$-th degree Taylor polynomials, defining a quasi-mean $M_f(a,b)=x_0$. For $f(z) = z^p$ and $r=3$, the real part of the pair of nonreal roots similarly yields a mean strictly between $a$ and $b$ (Horwitz, 2014).

5. Connections to Functional Equations and Smoothness

The QMVT equation is a functional-differential equation generalizing relations from classic analysis and mean-type function theory. In the symmetric case, differentiability and local regularity ($C^3$) are essential for deducing the ODEs that force the quadratic, exponential, or trigonometric solution forms. The earlier classification theorem by Balogh, Ibragimov, and Mityagin is subsumed as the special case $H(t)=t$ (Balogh et al., 2015). For Taylor polynomial means, smoothness of $f$ up to $C^{r+1}$ and sign-definiteness of $f^{(r+1)}$ are critical (Horwitz, 2014).

6. Corollaries, Partial Results, and Open Problems

A direct corollary is the exhaustive characterization of all $\varphi,\psi$ such that the Cauchy mean-value point can be chosen as a quasi-arithmetic mean: $$[\varphi(y) - \varphi(x)]\,\psi'\!\bigl( H^{-1} (\alpha H(x) + \beta H(y)) \bigr) = [\psi(y) - \psi(x)]\,\varphi'\!\bigl( H^{-1} (\alpha H(x) + \beta H(y)) \bigr)$$ holds for all $x < y$ if and only if $(\varphi,\psi)$ falls into one of the four described categories (Ibragimov, 2020). The Sahoo–Riedel problem, seeking all $(F, G, \phi, \psi)$ quadruples for which a general mean value identity holds at a fixed convex mean, is partially resolved using the QMVT classification (Balogh et al., 2015). For Taylor polynomials, extension to non-integer $p$ in the power means remains an open question with partial conjectural progress (Horwitz, 2014).

7. Broader Significance and Technical Implications

The Quasi-Mean Value Theorem provides the definitive structural description of when the mean point in the Cauchy MVT is not arbitrary but is given by an explicit, possibly nonlinear, mean. This has technical ramifications for the theory of functional equations, mean-value theorems, and means associated to Taylor expansions. It unifies and generalizes a spectrum of results in classical and modern analysis, bridging connections to the theory of means, the functional classification of differential equations, and the location of polynomial intersections (Ibragimov, 2020, Balogh et al., 2015, Horwitz, 2014).