Zdzisław Otachel

In this paper, some vector inequalities for convex maps are proved. The obtained results refer to the famous Jensen inequality and generalize further classical inequalities of Jessen and McShane. In addition, the Hahn–Banach theorems with sublinear and convex maps are considered and used to prove the theorem on the support of certain convex maps.

In this paper, versions of the famous Jensen inequality for sublinear isotonic functionals are proved. The obtained results generalize classic Jessen’s and McShane’s inequalities. Applications to generalized means and to Hölder’s and Minkowski’s inequalities are also given.

Summary: The theory of inequalities is an essential tool in all fields of the theoretical and practical sciences, especially in statistics and biometrics. Fujiwara’s inequality provides relationships between expected values of products of random variables. Special cases of this inequality include important classical inequalities such as the Cauchy–Schwarz and Chebyshev inequalities. The purpose of this article is to prove Fujiwara’s inequality in abstract probability spaces with more general assumptions for random variables than those associated with monotonicity as found in the literature. This newly introduced property of random variables will be called synchronicity. As a consequence, we obtain inequalities originated by Chebyshev, Hardy–Littlewood–Pólya and Jensen–Mercer under new conditions. The results are illustrated by some examples constructed for specific probability measures and specific random variables. In biometrics, the obtained inequalities can be used wherever there is a need to compare the characteristics of experimental data based on means or moments, and the data can be modeled using functions that are synchronous or convex.

The spraying procedure is one of the most difficult operations in agricultural production. Achieving the desired effectiveness of the procedure is dependent on obtaining an appropriate level and uniformity of liquid distribution. The aim of this paper was to present a liquid decomposition model generated on the basis of experimental data. The tests were carried out on a test stand, which consisted of a container with nozzles and a grooved table. The experiments were carried out with the use of selected standard, anti-drift, and air-induction single-stream nozzles at constant liquid pressure. The optimization process was carried out in Microsoft Excel Solver. Furthermore, in order to compare the data generated by the model with the data from the virtual boom, we applied an analysis of correlation and linear regression in the Statistica 13.1 software. Based on the results obtained, it can be concluded that the model is a good fit to the experimental data (R² > 0.95). The model, which was generated on the basis of experimental data, will facilitate control of the operation and degree of wear of nozzles, which will contribute to ensuring uniform spraying.

We present some new results on the Cauchy-Schwarz inequality in inner product spaces. Applications to reverse Bessel and Grüss type inequalities are given and a refinement of the Hadamard inequality is obtained. Sharpness of each introduced inequality is illustrated by suitable examples.