Mean Value Theorems and Functional Equations by P.K. Sahoo;T. Riedel

By P.K. Sahoo;T. Riedel

A complete examine suggest worth theorems and their reference to useful equations. along with the normal Lagrange and Cauchy suggest worth theorems, it covers the Pompeiu and Flett suggest price theorems, in addition to extension to better dimensions and the advanced airplane. moreover, the reader is brought to the sector of sensible equations via equations that come up in reference to the numerous suggest price theorems mentioned.

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Mean Value Theorems and Functional Equations

A complete examine suggest price theorems and their reference to useful equations. in addition to the conventional Lagrange and Cauchy suggest price theorems, it covers the Pompeiu and Flett suggest worth theorems, in addition to extension to better dimensions and the complicated airplane. moreover, the reader is brought to the sector of useful equations via equations that come up in reference to the various suggest worth theorems mentioned.

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29) where 6 is an arbitrary constant. 25), we see that h is an arbitrary function with a = h(0). Thus we obtain the solution as asserted in theorem for the case s = 0 = t. Case 2. Suppose s = 0 and t φ 0. 31) where a = h(0) and 6 = p(0). 32) for all x Φ y and x φ 0. 33), we see that /(0) = 6. 31) holds for all x in M. 5. Case 3. Suppose s Φ 0 Φ t. 34) for all real y φ 0 (where 6 = /(0)). 35) for all x φ 0 (where c = p(0)). 37) for all real nonzero x and y with tx Φ sy. 1. Suppose s = t. 37) yields x h{x) - y h(y) = (b - c)t + (x - y ) h(x + y).

First, we suppose x > 0 and let f(t) = (1 + i)n, for t G [0, x]. )n-1 > nx, and hence (1 + x)n > 1 + nx. The case when —1 < x < 0 can be handled similarly by considering f(t) = (1 + t)n for t G [x,0]. 3) with equality if and only if x = 1. To prove this, let f(t) = In i for t G [1, 6] and 6 > 1. The function / satisfies the hypothesis of the mean value theorem. )/'(» ,) Lagrange’s Mean Value Theorem and Related Functional Equations 32 which is and thus 1 — t < In 6 < 6 — 1. 4), we obtain Since 6 > 1, it follows that 0 < £ < 1.

13) transforms into x h(x) - y h(y) = (x - y ) h(x + y). 15) so also h + c, where c is an arbitrary constant. So we may suppose h(0) = 0. 16) that is λ is an odd function. 15), we get x h(x) - y h{y) = (x + y) h{x - y). 20) 42 Lagrange’s Mean Value. Theorem and Related Functional Equations for all u, v € R. Thus h{u) = au. 21) If we do not assume h(0) = 0, then we have in general h(u) = au + b. 14) this gives f(x) = x(ax + b) and, if we do not assume /(0) = 0, then f(x) = ax2 + bx + c. 12) are of the form f(x) = ax2 + bx + c h(x) = ax + b, where a, 6, c are arbitrary constants, as asserted.

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