Monotone and convex operator functions

Introduction.Monotone matrix functions of arbitrarily high order were introduced by Charles Loewner in the year 1934 [9](3) while studying realvalued functions which are analytic in their domain of definition, and continued in the complex domain, are regular in the entire open upper half-plane with non-negative imaginary part.The monotone matrix functions of order re defined later in the introduction deal with functions of matrices whose independent and dependent variables are real symmetric matrices of order re; monotone of arbitrarily high order means monotone for each finite integer re.The class corresponding to ra = 1 represents the functions which are monotone in the ordinary sense.Monotone operator functions are precisely the class of monotone matrix functions of arbitrarily high order.Loewner showed that analyticity plus the property of mapping the complex open upper halfplane into itself is characteristic for the class of monotone matrix functions of arbitrarily high order.Literature on this subject and its by-products, convex matrix functions and matrix functions of bounded variation, consists of but four papers, two by Loewner [9; 10 ] and one each from two of his doctoral students, F. Krauss [8] and O. Dobsch [5].The principal objective in this work was to discover whether convex operator functions (i.e.convex matrix functions of arbitrarily high order) satisfy properties analogous to those known for monotone operator functions.From the classical case it might be thought if a function is convex, its derivative should be monotone.Does this hold for operator functions?If not, what is true?It was deemed desirable to develop new representations for monotone operator functions.On physical grounds the interest in monotone operator functions is quite broad.A strong resemblance exists to the mathematical aspect of part of E. P. Wigner's work in quantum-mechanical particle interactions [20 ], the physical side being discussed by him in [18; 19].Connections with electrical network theory are mentioned by several writers [3; 4; 12].The Hamburger moment problem [16; 17] and the theory of typically real functions of order one studied by W. Rogozinski [15] and M. S. Robertson [13; 14] are both allied subjects.Even a slight relation with the special theory of relativity Presented to the Society, September 2,