By Washek F. Pfeffer
This ebook offers a close and more often than not simple exposition of the generalized Riemann-Stieltjes integrals found by means of Henstock, Kurzweil, and McShane. in addition to the classical effects, it comprises a few contemporary advancements hooked up with lipeomorphic swap of variables and the divergence theorem for discontinuously differentiable vector fields.
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Two functions a (x) and b(x) are said to be not comparable on an interval [c, d] if neither of the inequalities a (x) � b (x) nor a (x) :::: b(x) holds for all x E [c, d] . An inequality of the form a � b or a :::: b is said to be weak, whereas a < b and a > b are called strong. The classical results on inequalities are given in the books [BhB, HLP, Mit] . The methods used to derive inequalities vary considerably, and they often depend on the functions studied. Of the various methods and techniques used in this book, we can mention a few based on classical analysis: (a) Comparison of coefficients of power series.
Show that (a , n)(b, n) [ (d + c - a - b )ncd ( c, n + l )n ! (38) For lr l < 1 and a , b , c > 0 , c < (1 - ri F(a , b; c; r). Show that g ' (r) = (c - a)(c - b) a + b, (1 - let d =a+b-c rl- 1 F(a , b; c + 1 ; r). ab ]x n . and g (r) = Chapter 2 Gamma and Beta Functions We here recall the Euler gamma function r (z), which is the analytic continuation of the factorial function to C \ {O, -1, -2, . . }. This function is important in the sequel because many integrals can be written in terms of it.
2) If f is increasing and concave and f o g is convex, then g is convex. In particular, any positive log-convex function is also convex. (3) If f is decreasing and concave and g is increasing and convex, then f o g is decreasing and concave. (4) If f is positive, decreasing, and concave and g is positive, increasing, and concave, then g o f is decreasing and concave. Proof. For (1 ) let x , y be in the domain of g , and let a, b be positive numbers with a + b = 1 . : f(ag (x) + bg (y)) . : ag (x) + bg (y) , so that g is concave.