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**Symmetric properties of real functions.**
*(English)*
Zbl 0809.26001

Pure and Applied Mathematics, Marcel Dekker. 183. New York: Marcel Dekker, Inc.. xiii, 447 p. (1994).

The present book offers a detailed coverage of each important aspect of symmetric structures in functions of a single real variable. The ordinary difference \(\Delta f(x,t)= f(x+ t)- f(x)\) can be written as \(\Delta f(x,t)= {1\over 2} \Delta^ 1_ s f(x,t)+ {1\over 2} \Delta^ 2_ s f(x,t)\), where \(\Delta^ 1_ s f(x,t)= f(x+ t)- f(x- t)\) and \(\Delta^ 2_ s f(x,t)= f(x+ t)+ f(x- t)- 2f(x)\) are known as the first symmetric difference and second symmetric difference for the function \(f\). They represent odd and even increments of \(f\) at \(x\) and it is natural in some contexts to examine properties of \(f\) (e.g., the continuity or differentiability) by studying those properties in these two parts.

From author’s preface: “Chapter 1 begins the study by centering on the classical symmetric derivatives…we learn most of the basic properties of the first \((\lim_{t\to 0} \Delta^ 1_ s f(x,t)/2t)\) and second \((\lim_{t\to0} \Delta^ 2_ s f(x,t)/t^ 2)\) symmetric derivatives…Chapter 2 develops the continuity properties of functions satisfying some kind of symmetric growth condition…Taken together with Chapter 1 this might be considered as a brief introduction to the world of symmetric derivatives, symmetric continuity and like concepts. In Chapter 3 a variety of covering results are obtained that help to unify and clarify the arguments used in this subject. Chapter 4 begins a systematic study of the regularity properties of functions that satisfy some kind of even symmetric condition…Chapter 5 is devoted to monotonicity and convexity theorems as they arise from the first or second symmetric derivative. Chapter 6 does for the odd symmetry conditions what Chapter 4 has done for the even conditions…Chapter 7…We have defined a number of symmetric derivatives and this material attempts to survey the nature of the results so far known. Chapter 8 introduces the variational theory associated with the first and second symmetric derivatives. Chapter 9 concludes…with an account of the various symmetric integrals.”

The final part of the book contains a list of 49 open problems and a bibliography with 314 titles.

From author’s preface: “Chapter 1 begins the study by centering on the classical symmetric derivatives…we learn most of the basic properties of the first \((\lim_{t\to 0} \Delta^ 1_ s f(x,t)/2t)\) and second \((\lim_{t\to0} \Delta^ 2_ s f(x,t)/t^ 2)\) symmetric derivatives…Chapter 2 develops the continuity properties of functions satisfying some kind of symmetric growth condition…Taken together with Chapter 1 this might be considered as a brief introduction to the world of symmetric derivatives, symmetric continuity and like concepts. In Chapter 3 a variety of covering results are obtained that help to unify and clarify the arguments used in this subject. Chapter 4 begins a systematic study of the regularity properties of functions that satisfy some kind of even symmetric condition…Chapter 5 is devoted to monotonicity and convexity theorems as they arise from the first or second symmetric derivative. Chapter 6 does for the odd symmetry conditions what Chapter 4 has done for the even conditions…Chapter 7…We have defined a number of symmetric derivatives and this material attempts to survey the nature of the results so far known. Chapter 8 introduces the variational theory associated with the first and second symmetric derivatives. Chapter 9 concludes…with an account of the various symmetric integrals.”

The final part of the book contains a list of 49 open problems and a bibliography with 314 titles.

Reviewer: P.Kostyrko (Bratislava)

### MSC:

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

26A15 | Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable |

26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |

26A48 | Monotonic functions, generalizations |

26A51 | Convexity of real functions in one variable, generalizations |