In this paper, we respond to a critique of one of our papers previously published in this journal, entitled "TVOR: Finding Discrete Total Variation Outliers among Histograms". Our paper proposes a method for smoothness outliers detection among histograms by using the relation between their discrete total variations (DTV) and their respective sample sizes. In this response, we demonstrate point by point that, contrary to its claims, the critique has not found any mistakes or problems in our paper, either in the used datasets, methodology, or in the obtained top outlier candidates. On the contrary, the critique's claims can easily be shown to be mathematically unfounded, to directly contradict the common statistical theorems, and to go against well established demographic terms. Exactly this is done in the reply here by providing both theoretical and experimental evidence. Additionally, due to critique's compalint, a more extensive research on top outlier candidate, i.e. the Jasenovac list is conducted and in order to clear any of the critique's doubts, new evidence of its problematic nature unseen in other lists are presented. This reply is accompanied by additional theoretical explanations, simulations, and experimental results that not only confirm the earlier findings, but also present new data. The source code is at https://github.com/DiscreteTotalVariation/TVOR.

Abstract. Different approach to both Gautschi's inequalities (1) and (2) is given. This results in obtaining the best upper bound in (1) and the best lower bound in (2). The main result is the proof of the convexity of the function [Γ(x+t)/Γ(x+s)] 1/(t−s) for |t−s| < 1 . Several new very simple inequalities for digamma function, like ψ (x) < exp(−ψ (x)) or ψ (x + 1) < log(x + e −γ ) are also proved.Mathematics subject classification (1991): 33B15, 26D07.

Abstract. We study the asymptotics, box dimension, and Minkowski content of trajectories of some discrete dynamical systems. We show that under very general conditions, trajectories corresponding to parameters where saddle-node bifurcation appears have box dimension equal to 1/2, while those corresponding to period-doubling bifurcation parameter have box dimension equal to 2/3. Furthermore, all these trajectories are Minkowski nondegenerate. The results are illustrated in the case of logistic map.

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In this note it is proved that the integral arithmetic mean of a convex function is a Schur-convex function. Applications to Schur-convexity of logarithmic mean and gamma functions are given.