Zygfryd Kominek, a Mathematician, a Teacher, a Friend

For the first time I heard about a young mathematician of the name Zygfryd Kominek was in early 1970’s when I was a student of mathematics at the Silesian University in Katowice. I had friends studying physics at the same university, and one day I was told by one of them, that they had the calculus course with a guy who went in a military uniform. That was Zygfryd, who then performed military service, or rather a complement to it allowing him to be promoted to officer’s level. Not that he wanted it but that was in times of communist Poland where the military service was mandatory (students used to perform it during Summer holidays), and fresh PhD’s were proposed to make an additional service this was the case of Dr. Kominek. Later I met him personally, first indirectly (I used to have classes with Bożena Szymura


Introduction
For the first time I heard about a young mathematician of the name Zygfryd Kominek was in early 1970's when I was a student of mathematics at the Silesian University in Katowice. I had friends studying physics at the same university, and one day I was told by one of them, that they had the calculus course with a guy who went in a military uniform. That was Zygfryd, who then performed military service, or rather a complement to it allowing him to be promoted to officer's level. Not that he wanted it but that was in times of communist Poland where the military service was mandatory (students used to perform it during Summer holidays), and fresh PhD's were proposed to make an additional service -this was the case of Dr. Kominek. Later I met him personally, first indirectly (I used to have classes with Bożena Szymura who became Mrs. Kominek soon), and finally directly -we attended the same seminar on functional equations, founded and lead by Marek Kuczma.

Mathematician
The papers that were authored or co-authored by Professor Kominek range from functional equations and inequalities, real functions, measure theory, general topology, functional analysis, probability theory and stochastic processes, group theory and convex geometry.
Soon after our meeting at the Kuczma's seminar I was permitted to adress him as "Zyga", and we became close co-workers. Actually, I was one of the coauthors of only one paper, the other was Roman Ger (I mean the paper [19]). Let me quote a fragment of the review written by S. Łojasiewicz (cf. [MR1]) The authors generalize [M. Kuczma's and J. Smítal's, cf. [KS]] result in the following way: Let X be a metric space and µ * an outer measure satisfying some natural conditions (which are always true for R n and the Lebesgue measure). Let E, D ⊂ X, x 0 ∈ E and a map f : E ×X → X be such that f (x 0 ×D) is dense and f (x 0 , y) is an outer density point of f (E × y) in a uniform way with respect to y. Then µ * (B ∩ f (E × D)) = µ * (B) for every ball B of X. Several applications are given and a topological analogue is proved. Let us recall that the original Smítal's Lemma reads as follows.
Lemma 2.1. If E ⊂ R has a positive outer measure and D ⊂ R is a dense set in R, then the inner Lebesgue measure of the set R \ (E + D) is equal to zero.
The paper [19] was cited rather often. It contains (in the section "Backgrounds") several examples of applications of the [M. Kuczma's and] J. Smítal's Lemma to prove classical theorems in the theory of functional equations in several variables. For instance, the celebrated Steinhaus theorem, stating that the Minkowski sum of two measurable subsets of the real line contains a non-degenerate interval, can be proved (and actually is) with the help of the Smítal's Lemma. In a similar way the theorem of Fréchet and Sierpiński about the form of any measurable solution of the Cauchy equation, can be also derived from Smítal's Lemma. Also, the authors of [19] proved, using Smítal's Lemma, that every microperiodic and measurable function mapping reals to reals has to be constant almost everywhere, and each non-measurable subspace L of R over Q has to be saturated non-measurable (i.e. the inner Lebesgue measure of both L and R \ L are 0). Let me for instance quote the proof of Fréchet-Sierpiński theorem, which states that any measurable and additive function f : R → R is of the form f (x) = f (1)x, x ∈ R. Indeed, define g(x) := f (x) − f (1)x for every x ∈ R, and observe that g(r) = 0, r ∈ Q. Suppose that g = 0, and denote A := {x ∈ R : g(x) > 0}. Then −A = {x ∈ R : g(x) < 0} and 1 (A) = 1 (−A). Suppose that 1 (A∪(−A)) > 0. Then 1 (A) > 0, and we also have A = A+Q. From the Smítal's Lemma we infer that is of full measure. In particular, there is a y ∈ A ∩ (−A). We have g(y) > 0 and, since −y ∈ A we also have g(−y) = −g(y) > 0, which yields a contradiction. Therefore 1 (A) = 1 (−A) = 0, in other words g(x) = 0 a.e., say g(x) = 0 whenever x ∈ R \ W, where W is a nullset. As for an arbitrary z ∈ R also W − z is a nullset, we can find an h ∈ R \ (W ∪ (W − z)). Thus, taking into account additivity of g we get which ends the proof in view of arbitrary choice of z ∈ R.
These remarks make the sense of generalizing the classic Smítal's Lemma justified.

Single variable
Functional equations and inequalities are the main subject treated by Prof. Kominek. Among the papers cited by MathSciNet, we find about 40 devoted to the topic. Initially, he dealt with functional equations in a single variable, proving theorems on the existence and uniqueness of solutions of equations of the type or its generalization to a system where h i , f i,j , i, j ∈ {1, . . . , n}, are given and ϕ i , i ∈ {1, . . . , n} are unknown. In a number of papers, either written separately or jointly (with J. Matkowski) (cf. [4]- [6] and [8]- [13]) he studies the existence and uniqueness of convex solutions of (2.1) or the existence and uniqueness of solutions of class C r of (2.2), under some additional assumptions. In some cases the author proves also the continuous dependence of C r solutions of systems of the type (2.2). In papers [12] and [13] the author is interested in applying (his own) extensions of well known fixed point theorems to show the existence of solutions of a system of functional equations or to prove solvability criterion of systems of the type involving metric contractivity hypotheses of the type for all (x 1 , . . . , x n ) and (x 1 , . . . , x n ) from A = A 1 × · · · × A n , and all i ∈ {1, . . . , n}. Here we assume that absolute values of the eigenvalues of the matrix [a ik ] are less than 1.

Steinhaus and Piccard type results
Investigating functional equations in a single variable did not prevent Zygfryd Kominek from getting involved in the research concerning functional equations in several variables and related issues. Actually, the very first paper by him, which was published in prestigious journal Fundamenta Mathematicae (cf. [1]) concerns a topological counterpart of the celebrated Steinhaus theorem. It is not necessary to explain what is the importance of this theorem for development of the theory of functional equations in several variables. This way Kominek started his adventure with Steinhaus-type theorems, one of his specialities during the career. Here is a result from [1]: Theorem 2.1. Let X be a topological vector space over a topological field K. If A, B ⊂ X are second category Baire sets then A+B contains a nonempty open set.
From the result the well known theorem of S. Piccard (cf. [SP]) follows, and its version for Banach spaces was earlier proved by W. Orlicz and Z. Ciesielski (cf. [OC]).
In [1] the author mentions classes of sets connected with Jensen inequality or Cauchy equation. Investigation of such set classes is one of the leading subjects of Zygfryd Kominek research activity. For instance, in the paper [3] Kominek proves the following generalization of a result of Marcin E. Kuczma  In a number of papers, written by himself or co-authored, Zygfryd Kominek was busy with different aspects of sums or differences of sets. During the Ger-Kominek Workshop in Mathematical Analysis and Real Functions, held in Katowice in 2015, Władysław Wilczyński presented a talk dedicated to Kominek's achievements in this area (cf. [WW]). He mentioned in particular the following papers: [1], [3], [15], [16], [18], [19], [23] and [49]. Let me add to this list pretty general papers [28] and [68]. In [15] the author proves the following theorem. Proof. Take C -an arbitrary compact and nowhere dense set with positive Lebesgue measure. Put B := (Q − C) ∪ (Q + C). Then B ⊃ Q + C, and in view of the Smítal's Lemma B has infinite Lebesgue measure ((2)). Moreover, we have Q − B ⊂ B whence, if we put A := R \ B, follows (3). We also have (1), since B is a countable union of nowhere dense sets.
A generalization in a sense of the above theorem is the result proved in [16].
Theorem 2.4. Let f : R × R → R be such that f (x, ·) and f (·, y) are homeomorphisms for each real x, y. Then for any first category set A 0 with positive Lebesgue measure there exist subsets A and B of R such that In the same paper we find a generalization of a theorem by Harry I. Miller from [HM]. In early eighties of the last century both gentlemen met and started a fruitful cooperation. Harry visited Katowice and Zyga went to Sarajevo. They published two joint papers: [18] and [32]. In [18] the following theorem was proved.
Theorem 2.5. If A, B ⊂ R n , with A having a positive Lebesgue n-dimensional measure, and B having a positive n-dimensional outer Lebesgue measure then A + B contains an n-dimensional ball.
The proof is elementary and straightforward, but too long to be reproduced here. Let us note that the authors made a remark that their result is the best possible. Namely, if we assume that both A and B are non-measurable and have positive outer measure then Int(A + B) might be empty. Indeed, let us take an arbitrary Hamel basis H and let h 0 ∈ H be fixed. Let T be a subspace spanned by H \{h 0 } over Q. If we put C := T ∩(0, 1) then C is non-measurable with * 1 (C) = 1, which follows e.g. from [AA]. Setting A = B = C, we get that This is a measure theoretical version of the following topological theorem by Wolfgang Sander (cf. [WS1]).
Theorem 2.6. If A, B ⊂ R n , and A is of second category Baire set and B is of second category, then A + B contains an n-dimensional ball.
Yet another result by W. Sander (cf. [WS2]) is reproduced in [18] but an elementary and straightforward proof is given. In the paper [23] we find the following result, which we quote after a review due to H.I. Miller (cf. [MR2]): Assume that (1) (X; +) is a separable Baire topological group (not necessarily commutative); (2) µ is a measure defined on some σ-algebra M of subsets of X such that each subset of positive measure contains a compact subset of positive measure. The main result of the paper is the following theorem: Let assumptions (1) and (2) be fulfilled. Then the following conditions are equivalent: µ(A) > 0 and µ(B) > 0, then the set (−A + B) has nonempty interior. We remark that this very interesting theorem extends previously observed connections (see, e.g., [19]) between the classical Steinhaus theorem and Smítal's lemma in R n .
Zygfryd Kominek has been collaborating with many mathematicians. Some of them were already mentioned, and in connection with operations on sets he co-authored the paper [49] written jointly with W. Wilczyński. They proved, among others, the following.
It is worth to notice that the authors give an example showing that the assumption (2.3) is the best possible. Actually, they show that the set T ⊂ R p given by Let us mention here a result from [28], which concerns a purely algebraic situation, where instead of R or R n , one has a commutative group. Kominek states in [28] the following result.
As a corollary he gets the following. In the sequel of [28] Zygfryd is dealing with differences of subsets of commutative groups. It turns out that the situation is slightly different. In particular the following holds Theorem 2.11. Let A be an arbitrary subset of a commutative group G with the properties The paper deals also with a mixed case. We have Theorem 2.12. Let G be an arbitrary commutative group divisible by 2 and let A ⊂ G.
The paper [28] is concluded with yet another result.
Theorem 2.13. Let G be an arbitrary commutative group divisible by 2 and let A ⊂ G.
I am devoting pretty much place to the paper [28]. The reason for it is that in [28] the author shows his mastery, or even virtuosity of inventing the counterexamples, mostly based on specific constructions of Hamel bases. Zygfryd has been known for this skill, and I remember that during the seminars or international conferences he was answering questions just by producing a suitable example.

Classes of sets
Let us take a look on papers connected to some important classes of sets. We mean the papers [7] (written jointly with Bożena Kominek), [14] and [25], the latter written jointly with Roman Ger, Zygfryd's peer: they studied at the Jagiello University, Branch in Katowice. In [25] the following definition was recalled. If X is a linear topological Baire space then B(X) = {T ⊂ X : every additive functional defined on X (2.4) and bounded from above on T is continuous}.
In the same paper we find the following definition. The definition of both classes was introduced by R. Ger and M. Kuczma in [GK]. In [MEK] Marcin E. Kuczma proved that for each n ∈ N. In the paper [25] the authors prove the following theorem, in a completely different way than that given by M.E. Kuczma in [MEK]. Actually, they had to elaborate a new method of proving the result, because M.E. Kuczma's method relied on the finite dimension of the domain (see the comment below).
Theorem 2.14. The equality A(X) = B(X) holds for any real linear topological Baire space.
While M.E. Kuczma to prove (2.6) used mainly a generalization of the fundamental Hahn-Banach extension theorem, Ger and Kominek apply rather results by Gerd Rodé from [GR] and a refinement of a theorem by Norbert Kuhn (cf. [NK]).
In the papers [7] and [14] the authors introduce the following classes of subsets of R n , with n = 1 in the case of [7], for any open and convex set D ⊃ T every Jensen convex function f : In both papers relations between classes A(R n ), B(R n ), A n C , and B n C are studied. In [7] the authors give examples which show that for n = 1 Thus we see that a slight change of the definition implies rather big differences between classes. In the paper [14] Kominek describes properties of classes A n C and B n C . In particular he proves the following.
Theorem 2.16. If g : [a, b] → R is continuous and not affine then the set In other words continuity of the restriction of a Jensen convex function f : D → R where R 2 ⊃ D to the graph of a non-affine continuous function g : [a, b] → R implies it continuity. In the proof uses the fact that I g is compact and f has to be upper bounded on the set 1 2 (I g + I g ) which has a positive measure (see [MK]). It follows that f is continuous.

Semilinear topology
In 1980's Zygfryd was visiting rather often Marek Kuczma who was then very ill from the physical point of view but extremely healthy mentally. Kuczma was very active mathematically, and Kominek was co-author of four papers at the time. These were papers [26], [30], [36] and [39]. It is natural that we join here the paper [31] which was actually the habilitation dissertation of Zygfryd. The basic idea which appears in [26] is the concept of the semilinear topology. Let X be a linear space over a field K (Q ⊂ K ⊂ R) endowed with a topology T . Consider the function ψ : K × X × X → X defined by ψ(λ, x, y) = λx + y for every λ ∈ K and x, y ∈ X. If the function ψ is continuous with respect to T then we say that the topology T is linear, if it is separately continuous, we say that T is semilinear. The authors of [26] prove that T is semilinear if, and only if, for every fixed λ = 0 and y ∈ X the function γ : X → X given by is continuous, and for every fixed x, y ∈ X the function ω : K → X given by ω(λ) = λx + y, is continuous at 0. An example of semilinear topology is the so called core topology. The core topology is the topology of algebraically open subsets of X, i.e. such sets A ⊂ X that A = coreA. By coreA we understand the set of all points y ∈ A which are algebraically interior to A. A point y ∈ A is said to be algebraically interior to A if for every x ∈ X there exists an ε > 0 such that for every λ ∈ (−ε, ε) ∩ K. Recently, the coreA is called algebraic interior of A with respect to K and denoted by algint K (A). If K = R we simply write algint(A).
Denote by T (X) the core topology in the space X over K. It turns out that T (X) is semilinear, and that for any semilinear topology T we have T ⊂ T (X). The inclusion is strict if dimX > 1. The concept of the core topology was very inspiring for many colleagues working in the functional equations theory. Zygfryd himself is the author of a brilliant result stating that a midconvex functional is convex if and only if it is continuous in the core topology. Actually this is a formulation remembered by his colleagues, an abbreviation expressing the great idea. This follows for instance from the following results contained in [31] (Theorem 3. 1 and Lemma 3.7, cf. also [26] and [30]). Let us recall that Kominek deals with the case where K = R.
Theorem 2.17. Let X be a linear space (over R) endowed with a semilinear Baire topology (hence in particular with the core topology), let D ⊂ X be an open and convex set, and let f : D → R be a Jensen convex function. If f is lower semicontinuous in D then it is continuous in D.
Without the assumption on Baire property, Zygfryd obtained the following (cf. Lemma 3.7 in [31]). The ideas connected with the core topology were inspiring for other authors. Let us mention the paper by K. Baron, M. Sablik and P. Volkmann [BSV], where continuity with respect to the core topology is a condition equivalent to the "decency" 1 of a character of a rational linear space. Actually we proved the following. We say that a function f : D → R, where D is a convex subset of a linear space X, is Wright-convex if it satisfies the following inequality for every x, y ∈ D and λ ∈ [0, 1]. In [31] Kominek proved the following nice result, extending some earlier one obtained by C.T. Ng in [NG] where the final dimensional case is considered. for every x ∈ D. 1 A character χ : G → T of a group G is "decent" if it has the representation χ(x) = exp(2πif (x)), x ∈ G, with an additive f : G → R.
In [31] several results concerning classes A(X, T ) and B(X, T ) which are defined analogously to the classes given by (2.5) and (2.4), only here the notion of openness and continuity refer to the semilinear topology T . In particular, the equality is proved in [31] under the assumption that (X, T ) is a Baire space. The result extends further the classical result by Marcin E. Kuczma (cf. [MEK] and [25], the papers cited above). Besides, in [31] we find extensions of several classical results, such as Bernstein-Doetsch and Mehdi theorems. Let us mention also paper [43], as example of description of additive functions continuous with respect to the algebraic topology.
Let us mention here a student of Zygfryd, Andrzej Olbryś who has been specializing in different variations of the classical convexity for many years. In particular, he got interested (or rather was advised to show interest) in the Wright-convexity by Zygfryd, while preparing his PhD dissertation. Andrzej Olbryś uses in his work the notion of algebraically open subsets of a linear space. Let us mention for instance the paper [AO] where the author deals with sets D which have nonempty coreD with respect to Q(t), t is a fixed number from (0, 1), (an extension of the field Q by the element t) or with respect to R. Olbryś has extended the notion of convexity to the so called (ω, t)-convexity.
if and only if ω satisfies the conditions (a)-(c) for some point y ∈ algint Q(t) (D).

Stability
Among the papers of Kominek, there is a number devoted to the questions of stability, i.e. problems asked originally by S. Ulam with respect to the additive functions, and first solved by D. Hyers, but then extended to a separate theory. Hyers and Ulam probably would not recognize the original question in the massive flood of results which multiply each year. And Zygfryd was here at the beginning of this brilliant career of the stability of functional equations and inequalities. Let us quote [29], [44], [45], [52], [58], [60], [61], [64], [65], [72], [76], and, to some extent, also [57].
Let us begin with recalling original question of S. Ulam (cf. [DH] and [HU]).
Problem 1. Let E and E be Banach spaces and let δ be a positive number. A transformation f (x) from E to E will be called δ-linear if f (x+y) − f (x) − f (y) < δ for all x and y in E. Then the problem may be stated as follows. Does there exist for each > 0 a δ > 0 such that, to each δ-linear The question was answered in affirmative by D.H. Hyers in [DH]. The question was then generalized in several directions, since this is not a survey of the vast area of stability we will not present all the issues. Let us concentrate on some selected results obtained by Zygfryd. In [65] the following theorem was proved.
Theorem 2.21. Let S be an abelian semigroup, X be a Banach space, and let g : S → X be a function satisfying the following inequality where ω : S * × S * → R fulfils conditions Then there exists a unique function f : S → X satisfying equation and the estimation where ϕ : S * × S * → R is given by Let us note an interest in stability of the so called Jensen-Hosszú functional equation for all x, y ∈ R. He proved in [67] the following.
Theorem 2.22. Let δ > 0 be a real constant and let g : R → R be a function satisfying the following inequality for all x, y ∈ R. Then there exists an additive function a : R → R such that for every x ∈ R.
Note that the equation (2.7) arises when we take left-hand side of the Hosszú equation and compare it with the right-hand side of the Jensen equation which hold in any set where the inside operations have sense. Both (2.9) and (2.10), in case of any real interval have the same solutions: f (x) = a(x)+b, where a : R → R is an additive function, and b ∈ R is a constant. Thus Theorem 2.22 expresses the fact that (2.7) is stable when treated for functions mapping reals to reals. Another natural domain for solutions of (2.9), (2.10) and (2.7) is the interval I ∈ {(0, 1), [0, 1]}. In this case however the behaviour of (2.10) and (2.9) is slightly different. Despite the fact that they remain equivalent, their stability properties differ. Actually, (2.10) remains stable in the sense of Hyers-Ulam, but (2.9) is not stable anymore (it was shown by Jacek Tabor in [JT]). Zygfryd, together with Justyna Sikorska wrote the paper [70], in which they proved the following.
Using a slightly different method (cf. [70]) they were able to prove also the following.
Kominek continued the subject in [77], proving results for "pexiderized versions" of the Hosszú-Jensen equation. In particular, there is an example in [77] showing that in the most general case, i.e. inequality 2f x + y 2 − g(x + y − xy) − h(xy) ≤ δ which is fulfilled by functions f, g, and h mapping the unit interval I into R, yet they are not "close" to affine mappings, as it could be expected. The positive results are obtained in cases where we restrict the number of functions to two (see Theorems 1-3 in [77]). Let us also mention the paper [60], where the question of stability of the conditional Fréchet equation was studied (for increments from a subset C of a semigroup X).
Zygfryd wrote also a number of papers devoted to stability of inequalities. We have in mind the papers [21], [35], [61], [62]. For instance in [35] we find the following theorem.
Theorem 2.25. Let (X, +) be a commutative, 2-divisible group. If f : X → R is an ε-subadditive function (i.e. the inequality f (x + y) ≤ f (x) + f (y) + ε is satisfied for all x, y ∈ X) then there exists a Jensen function h : X → R such that The main tool of the proof is a theorem of R. Kaufman from [RK] on the existence of an additive function separating superadditive (m) and subadditive (M ) ones, all defined in a commutative semigroup, and such that m ≤ M.
We say that 2-divisible topological group X has the property ( 1 2 ) if and only if for every neighbourhood V of zero there exists a neighbourhood W of zero such that The main result reads as follows.
Theorem 2.26. Let X be a 2-divisible topological abelian group having the property ( 1 2 ), which is generated by any neighbourhood of zero in X 2 . Assume that a subquadratic function ϕ : X → R satisfies the following conditions: (i) ϕ(0) ≤ 0; (ii) ϕ is locally bounded from below at a point of X; (iii) ϕ is upper semicontinuous at zero. Then ϕ is continuous everywhere in X.
In the proof of Theorem 2.26 the authors in particular use the following observation.
Observation 1. Let X be a 2-divisible topological group having the property ( 1 2 ), f : X → R be an arbitrary function. Then for every u ∈ X, each ε > 0 and every neighbourhood W 0 of 0 ∈ X there exists a neighbourhood W of 0 ∈ X such that 1 2 W ⊂ W ⊂ W 0 and The same Observation appears in [73] as Lemma 1. It serves, among others, to prove the main result from the paper.
Theorem 2.27. Let ϕ : X → R be a function satisfying the so called Drygas inequality If ϕ is locally bounded below at each point of X and upper semicontinuous at zero, then it is continuous in X.
Theorem 2.27 implies then a series of other results stating the continuity of ϕ which is bounded below only at points x 0 = 0 and −x 0 (Theorem 2 in [73]) or is continuous at 0 (Theorem 3 in [73]) or it is semicontinuous at 0 and bounded below at a point x 0 = 0 (Theorem 4 in [73]).
We cannot refrain from mentioning the unique paper written jointly with our late colleague Zbigniew Gajda (cf. [37]). In that paper they replace commutativity or amenability of a semigroup S by its weak commutativity i.e. the condition stated as follows.
Definition 2.28. We say that the semigroup (S, ·) is weakly commutative, iff Let (S, ·) be a semigroup and consider functions f, g : S → R satisfying the following conditions. (2.14) for every x, y ∈ S. The notion of weak commutativity has been used by several authors but Gajda and Kominek proved in particular the following.
Theorem 2.29. Suppose that the semigroup (S, ·) is weakly commutative and let f, g : S → R satisfy (2.12), (2.13) and (2.14). Moreover, assume that Then there exists a unique additive function a : S → R such that for every x ∈ S.
They proved also another theorem using the assumption of weak commutativity, instead of (2.15) assuming that S is a group.
Instead of assuming that S is a group it is possible to admit that S is amenable 3 . The authors of [37] showed the following.

Alienation
In the paper [78] together with Justyna Sikorska, Zygfryd proved a theorem on alienation of the logarithmic and exponential equations. They considered the equation In particular, they proved the following.
They asked in [78] the following open question.
The Problem was answered by Gyula Maksa in [GM]. He proved the following

Teacher and friend
Zygfryd Kominek was born on July 30th, 1945 in Radlin, a town south of Katowice. He attended schools there and in 1963 he started his studies in mathematics at the branch of Jagiello University in Katowice. Here he obtained master's degree in 1968, and began his employment after the Summer holidays in 1968 -but it was already the brand new Silesian University. He received his PhD degree in 1974, and his supervisor was Bogdan Choczewski from Cracow. In 1991 he got the habilitation from the Warsaw University of Technology. Since 2008 Zygfryd Kominek is full professor in mathematics.
Professor Kominek has been working in the Silesian University for almost 50 years. In the meantime, he was employed in some other academic institu- At the Institute of Mathematics of the Silesian University, Zygfryd Kominek served as the director (1996)(1997)(1998)(1999)(2000)(2001)(2002) and deputy director (1992-1996 and 2012-2015), he was also a deputy chairman of the Publishing Committee of the Silesian University and a representative of the faculty in the Senate of the University. I have to underline in particular his service as deputy director from 2012 till 2015. He was then responsible for research in the Institute, and I, as the director of the Institute, owe him a lot. Kominek was also the Editor in Chief of Annales Mathematicae Silesianae.
He was leading the prestigious "Tuesday" Seminar on Functional Equations for several years (from October 2009 till June 2015). The seminar was founded in 1963 by Marek Kuczma, and has been continued, meeting every Tuesday of academic year, at 4:15 PM.
Zygfryd is also a teacher. He has five students, chronologically: Tomasz Zgraja, Jacek Mrowiec, Andrzej Olbryś, Michał Lewicki and Katarzyna Troczka-Pawelec. One of them, Andrzej Olbryś, has received his habilitation in 2019. He has joint papers with three of them, and Andrzej Olbryś has been under his influence ever since defending the PhD. His habilitation is actually composed of results touching in general the different kinds of convexity and creatively developing the concepts of "sandwich theorems" and supporting functionals.
Zygfryd took part in numerous conferences on functional equations and inequalities, and real functions, usually giving talks on his own results and presenting remarks and problems that inspired many mathematicians. In the present volume you will find a paper by Harald Fripertinger who answered a Kominek's question asked during the 54th ISFE held in Hajdúszoboszló in Hungary, in 2016 (cf. [RI]).
For many years I participated in the "Monday" seminar on functional equations and inequalities in several variables, founded in 1982 and lead continuously till 2016 by Roman Ger. Since October 2016 I became the leader of the seminar. When I open the meeting, on my right, close to the window, I can see two friends: Roman and Zyga, with whom I wrote the paper [19]. They are still attending the meetings despite the fact that for some time they are both emeriti. Still, their comments and remarks are extremely valuable and I am disappointed indeed, if for some reasons one of them is missing. It is hard for me to imagine the seminar without both of them. We organized in 2015 a special conference, named Ger-Kominek Workshop in Mathematical Analysis and Real Functions, dedicated to the two friends on the occasion of their seventieth anniversaries. More than a hundred of mathematicians came from all over the world and congratulated our colleagues. Though the celebration was great, it did not mean the crowning of their mathematical activity, which has gained significant momentum at that time.
We wish you, Dear Zygfryd, many successes in future, many new results, and many meetings with friends! Zygfryd Kominek: List of publications