By Isaac Todhunter

This Elibron Classics publication is a facsimile reprint of a 1864 version by means of Macmillan and Co., Cambridge and London.

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**Example text**

Y, y) (13) . AppIying the same inequality, this time for the nonnegative funetion K(x, y) and the eoneave funetion v(x), we obtain the end result that 1 { Ju(x)v(x)dx~min min O y . (z, z) JJ . (14) It remains to show that this inequality is nontrival. Adireet ealeuIation, earried out by O. GROSS, shows that the minimum is attained at y = 1, z = 0 and at the symmetrie point y = 0, z = 1, yieIding the vaIue 1/2. The minimal funetions are as stated. § 41. 7J. The proof we give is due to WEINBERGER.

Introduction As soon as we leave the well-traversed fields of real and complex numbers for the broader and relatively unexplored domains of hypercomplex numbers, we open the way for the introduction of many different types of ordering relationships. In this chapter, we shall discuss a variety of interesting inequalities centering about the theme of matrices. As we shall see, the basic concept of positive number can be extended to matrices in many different and significant ways. The simplest and most immediate extension is the notion of positive definite matrix.

A) 87, 225-229 (1912). § 16. The fact that M,(x, IX) is not necessarily a convexo-concave function was pointed out in 1. : On the convexity of mean value functions. Bull. Am. Math. Soc. 54,770-776 (1948). The inequality st1(x) ;;;; St. (x) for 0 < t1 < t. was established by ]ENSEN in 2. JENSEN, J. L. W. : Sur les fonctions convexes et les inequalites entre les valeurs moyennes. Acta Math. 30, 175-193 (1906) as an application of his new theory of convex functions. As ] ensen stated, however, the inequality had previously been proven by PRINGSHEIM in 3.

### A Treatise On The Differential Calculus with numerous examples by Isaac Todhunter

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