Xiong Qinglai presented his main thesis Sur les fonctions entières et les fonctions méromorphes d'ordre infini and his second thesis La théorie de Lebesgue et ses applications fondamentales to the Faculty of Science, Paris, in 1934. It was examined by E Borel (President), A Denjoy and G Valiron and approved on 29 June 1934.

1.1. From the Introduction.

Since the work of H Poincaré and J Hadamard, especially after the discovery of the famous theorem of E Picard in 1879 and its elementary demonstration by M E Borel in 1896, the theory of integer functions has motivated the research of many geometers. We can distinguish two kinds of results: the first are of a qualitative nature like those of Montel, Julia, Ostrowski, etc.; the second of a quantitative nature like the theorems of Borel, Schottky, Landau, Blumenthal, Denjoy, Valiron, Milloux, etc.

From this last point of view, it is above all thanks to the notion of order introduced by Borel that many results can take a precise and simple form. With this notion, Borel gave a very precise theorem concerning the density of zeros of an integer function of finite order, contained in a given circle with centre O, a theorem which is too well known to be recalled here. It was by seeking an analogous proposition in the study of the distribution of arguments that, later, G Valiron introduced what he calls Borel's direction.

In the case of functions of infinite order, Borel did not explicitly define the orders, and his results given in his Fundamental Memoire of Acta Mathematica, concern only a class of functions of infinite order, but he laid the foundations of a general theory and, by a powerful intuition, he recognised the difficulties of the problem and indicated the principal means of surmounting them.

Inspired by the ideas of Borel and completing his results, Blumenthal succeeded, by means of standard functions, in constructing a theory comprising all integral functions without exception. But his results are not as tight as those obtained by Borel in the case of finite order and in the case of functions of infinite order which can be treated by his method.

With regard to the canonical products, A Denjoy, with the help of an assumption made on the mode of the growth of the zeros, obtained for the maximum modulus a very tight limitation.

The fruitful method of logarithmic mean values   created a few years ago by R and F Nevanlinna has given many results a definitive form and makes it possible to deal with a large number of problems with precision and simplicity.

The main purpose of this work is to redo the general theory of integer functions of infinite order to seek to obtain better results than those of Blumenthal and to extend it at the same time to the more general case of meromorphic functions of infinite order; the degree of precision in view is that included in the results given by Borel in the case of finite order.

1.2. Acknowledgement.

It remains for me, to end this introduction, to express all the gratitude I have contracted towards MM Borel and Hadamard. I particularly thank M Valiron who suggested this work to me and whose indications and criticisms were very precious to me, and M Denjoy who was also good enough to give me advice. Allow me once again to express my sincere gratitude to MM H Villat, P Montel and P Humbert. Nor can I forget those who helped me and made my stay in Paris possible.

1.3. Hand written by the author on the copy.

"A Monsieur le Professeur Montel et cher Maître Hommage respectueux. K L Hiong"

3.1. Review by: S-N Patnaik.
zbMATH 04075010

This volume contains a collection of 16 papers by the Chinese mathematician King-Lai Hiong on function theory, especially on meromorphic functions, a bibliography of his publications, and a preface giving a glimpse of his life as an outstanding mathematician and educationalist. Hiong had spent several years studying and doing research on function theory in France and towards the later part of his life, he was the Director of the Division of the Theory of Functions at the Institute of Mathematics of the Chinese Academy of Sciences. Several of the papers in this volume continue and to some extent overlap the important work of the leading function-theorists such as Hadamard, E Borel, P Montel, and Nevanlinna and others. The contributions contained in this volume rank Hiong as a function-theorist of the first rank even comparable with E Borel and Nevanlinna. One of his papers treats algebraic functions and their derivatives in which he uses the methods originally used by E Borel and H L Selberg. There are important papers on entire functions, meromorphic functions of infinite order, and generalisations of the fundamental theorems of Nevanlinna. Finally, this volume should be of great interest to research workers and mathematicians interested in function theory and meromorphic functions.

4.1. From the Introduction.

We know how important is the role played by the logarithmic method due to R Nevanlinna, in the modern theory of meromorphic functions. The second fundamental theorem, which constitutes a powerful tool, has given rise to numerous extensions followed by various applications. In the present fascicule, we do not concern ourselves with the result of T Shimuzu, nor with the theory of L Ahlfors which are based on geometric considerations. We confine ourselves to more direct extensions and only for meromorphic functions in the whole open plane or in a circle; but we reserve an important place for what concerns the theory of algebroid functions and that of systems of functions.

We begin by presenting extensions obtained by involving derivatives. A fruitful idea of P Montel is at the origin of this kind of research. It was on his suggestion that G Miranda arrived, after partial results found by F Bureau, to establish the theorem known by his name; and in the same order of ideas, H Milloux, while seeking a basis for the theory of meromorphic functions in relation to their derivatives, obtained various fundamental inequalities. But it is worth mentioning a result apart from E Ullrich in which the derivative of the function considered also intervened. The first results of Milloux are already to be found in a statement made by himself; we give here only those he obtained later as well as results due to other authors.

By introducing a supposedly meromorphic primitive, we were able to establish a simple inequality which also extends the second fundamental inequality and which is well suited for applications. We think it is useful to also give this result here.

Then, we approach the extension of the logarithmic method to algebroid functions. Although the first results were obtained by H L Selberg and by G Valiron more than 20 years ago, there is still, to our knowledge, no collection containing this already well developed theory. We particularly expose the method of Valiron; to complete what concerns his second fundamental theorem, we carry out the proof, for which he indicated only the identity which can be used as a starting point. We are also adding a new extension.

With regard to the theory of systems of functions, an extension of the second fundamental theorem was first given by R Nevanlinna himself with interesting consequences, and by the same method H Cartan made another followed by applications from the finite point of view. Then, by means of another method due to L Ahlfors, H Weyl and J Weyl developed a theory of meromorphic curves and, retaining the denomination of these two authors, but employing a different method, Ahlfors succeeded in constituting, for systems of functions, a more complete theory, which however has not received applications. We do not expose these last two theories and we briefly give the main results provided by the method of R Nevanlinna. We devote the last part of our presentation especially to another theory, that of H Cartan on the linear combinations of p holomorphic functions. There are several applications; in particular, for an important class of algebroids, we deduce from this theory a more precise fundamental inequality than that previously established.

For the main theorems contained in this presentation, we endeavour to give sufficient indications so that the reader can reconstruct the complete proofs without great difficulty.

In closing, I would like to express my sincere gratitude to Henri Villat who kindly asked me to write this booklet for his beautiful collection, the Mémorial des Sciences Mathématiques.

4.2. Review by Olli Lehto.
Mathematical Reviews MR0091341 (19,950b).

In this monograph, classical problems of Nevanlinna's value distribution theory for meromorphic functions are treated by means of purely analytical tools. It is assumed that the reader possesses a certain acquaintance with the problems and methods of this theory.

Applying the standard methods of Nevanlinna theory and making use of results by Montel, Miranda, Milloux and others, the author studies, in the first chapters, relations between the distribution of values of a meromorphic function and its derivatives. Because of the character of the problems posed, the results in this direction are often rather incomplete and formally complicated. In spite of this, the author has succeeded in finding some interesting new results.

After this, value distribution properties of algebroid functions are dealt with. More than twenty years have now passed since a fairly systematic theory was built up, primarily by H Selberg and Valiron, but no unified representation has been published before. It is, therefore, well motivated to include this theory in the book, although this chapter contains little new material.

The last part of the monograph is devoted to the study of linear combinations of analytic functions. ... the chief attention is concentrated on the theory developed by H Cartan, which is reviewed in fairly great detail.