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Classroom notes
References [1] Salas, S., Hille, E. and Etgen, G., 2003, Calculus: One Variable, 9th edn (New York: John Wiley and Sons, Inc.), pp. 611–614, 616. [2] Spivak, M., 1994, Calculus, 3rd edn (Houston, TX: Publish or Perish, Inc.), pp. 201–202, 210–211. [3] Stewart, J., 1999, Calculus, 4th edn (Pacific Grove, CA: Brooks/Cole Publishing Company), pp. 486–487. [4] Thomas, G. (revised by M. Weir, J. Hass, and F. Giordano), 2006, Thomas’ Calculus (Early Transcendentals), 11th edn (Reading, MA: Addison Wesley), pp. 316–320.
More notes on a functional equation FENG QI*y, JIAN CAOz and DA-WEI NIUz yResearch Institute of Mathematical Inequality Theory, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China zSchool of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China (Received 10 August 2005) This note further discusses the functional equation f ðxyÞ ¼ xf ðyÞ þ yf ðxÞ and discusses two generalizations of it.
1. Introduction The functional equation f ðxyÞ ¼ xf ðyÞ þ yf ðxÞ
ð1Þ
was studied by [1–3] and subsequently by [4]. The general solution is f ðxÞ ¼ Ax ln jxj
ð2Þ
with f ð0Þ ¼ 0 if f (x) is assumed continuous, although there are other solutions that are everywhere discontinuous [5]. We offer an alternative derivation of this result and discuss two generalizations.
*Corresponding author. Email:
[email protected];
[email protected];
[email protected]
[email protected];
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Classroom notes
2. An alternative approach to equation (1) For non-zero x and y, rewrite equation (1) in the form f ðxyÞ f ðyÞ f ðxÞ ¼ þ xy y x
ð3Þ
If we now define ðxÞ ¼
f ðxÞ x
ð4Þ
we reach the multiplicative Cauchy functional equation ðxyÞ ¼ ðxÞ þ ð yÞ
ð5Þ
whose general solution (in continuous functions) yields equation (2). Again, however, discontinuous solutions exist [6, 5], and if !(x) is any discontinuous solution of equation (5), then f ðxÞ ¼ x!ðxÞ is a solution of equation (1).
3.
Generalizations of the functional equation (1)
It is easy to see that the following two functional equations involving n variables are generalizations of equation (1): f
n Y
! xi
¼
n X
i¼1
f
n Y
Y
xk f
xi
¼
X
i¼1
xi
ð6Þ
i6¼k
k¼1
!
!
¼ 1n
Y
k
! xi f ðxk Þ
ð7Þ
i6¼k
where ðx1 , x2 , . . . , xn Þ 2 Rn and n 2. If setting n ¼ 2, then (6) and (7) are equivalent to (1). The functional equations (6) and (7) have a trivial solution f ðxÞ 0 for x 2 R clearly. If xi 6¼ 0 for all 1 i n, then (6) and (7) can be rewritten respectively as f
Q
n i¼1 xi Qn i¼1 xi
¼
n f X
Q Q
k¼1
i6¼k
i6¼k
xi
xi
ð8Þ
and f
Q
n i¼1 xi Qn i¼1 xi
¼
n X f ðxk Þ k¼1
xk
ð9Þ
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Classroom notes
If xi 6¼ 0 for all 1 i n and setting f ðxÞ ¼ xgðln jxjÞ for x 6¼ 0, where g(x) is a new unknown function, then (6) and (7) can also be rewritten respectively as g
n X
! lnjxi j ¼
i¼1
n X
X
g
! lnjxi j
ð10Þ
i6¼k
k¼1
and g
n X
! lnjxi j ¼
i¼1
n X
gðlnjxk jÞ
ð11Þ
k¼1
For n 2, letting x1 ¼ x2 ¼ ¼ xn ¼ 1 in (6) yields f ð1Þ ¼ n f ð1Þ
ð12Þ
Hence, f ð1Þ ¼ 0. For n 3, setting x2 ¼ x3 ¼ ¼ xn ¼ 1 and x1 ¼ x 2 R, then it follows from (6) that f ðxÞ ¼ ðn 1Þf ðxÞ
ð13Þ
Thus, f ðxÞ ¼ 0 for x 2 R. This means that the functional equation (6) for n 3 has the unique solution f ðxÞ 0, which is continuous but trivial. Now it is clear that (9) and (11) are trivially inductive generalizations of gðln jxj þ ln jyjÞ ¼ gðln jxjÞ þ gðln jyjÞ
ð14Þ
ðx þ yÞ ¼ ðxÞ þ ð yÞ
ð15Þ
whose standard form is
and (3) respectively, so we can say that (7) is a trivially inductive generalization of (1). Further, it is not difficult to verify that these functional equations have a nonzero continuous solution f ðxÞ ¼
x loga x, 0,
x 6¼ 0 x¼0
ð16Þ
for a > 0 and a 6¼ 1.
Acknowledgements The authors would like to express their sincere thanks to the anonymous referee for his/her valuable comments and detailed modifications on this note. The first author was supported in part by the Science Foundation of Project for Fostering Innovation Talents at Universities of Henan Province, China.
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Classroom notes
References [1] Ren, Zh.-P., Wu, Zh.-Q., Zhou, Q.-F., Guo, B.-N. and Qi, F., 2004, Some notes on a functional equation. International Journal of Mathematical Education in Science and Technology, 35, 453–456. [2] Zhang, Sh.-Q. and Qi, F., 1996, On some generalizations of a mathematical proposition. Academic Forum of Nandu (Journal of Nanyang Teachers’ College), 16, 65–66. (in Chinese) [3] Zhou, Q.-F., Wu, Zh.-Q., Guo, B.-N. and Qi, F., 2003, Notes on a functional equation. Octogon Mathematics Magazine, 11, 507–510. [4] Deakin, M.A.B., 2006, More on a functional equation. International Journal of Mathematical Education in Science and Technology, 37, 246–247. [5] Smital, J., 1988, On Functions and Functional Equations (Bristol: IOP). [6] Acze´l, J., 1966, Lectures on Functional Equations and their Applications (New York: Academic Press).
On the weighted mean value theorem for integrals M. POLEZZI* Universidade Estadual de Mato Grosso do Sul-(UEMS) Rodovia MS 306, Km 6, Cassilaˆndia, Brazil (Received 6 July 2005)
1. Introduction The Mean Value Theorem for Integrals is a powerful tool, which can be used to prove the Fundamental Theorem of Calculus, and to obtain the average value of a function on an interval. On the other hand, its weighted version is very useful for evaluating inequalities for definite integrals. Mean Value Theorem for Integrals: Let f: [a, b] ! R be a continuous function. Then, there exists a number c 2 (a, b) such that Z
b
fðxÞdx ¼ fðcÞðb aÞ a
Thus, if x 2 (a, b), it is possible to choose a number a < cx < x as a function of x on (a, b) such that Zx fðtÞdt ¼ fðcx Þðx aÞ ð1:1Þ a
*Email:
[email protected]
