We construct and analyze Bernstein-type operators on triangles with curved sides, their product and Boolean sum. We study the interpolation properties and approximation accuracy. Using the modulus of continuity we also study the remainders of the corresponding approximation formulas. Finally, there are given some particular cases and numerical examples.

We prove that the partition function p(n) is log-concave for all n > 25. We then extend the results to resolve two related conjectures by Chen and one by Sun. The proofs are based on Lehmer’s estimates on the remainders of the Hardy–Ramanujan and the Rademacher series for p(n).

In the paper, Binet’s first formula for the logarithm of the gamma function is extended, inequalities and star-shaped and sub-additive properties of the extended remainder of Binet’s first formula for the logarithm of the gamma function are presented, and completely monotonic properties of the difference between extended remainders of Binet’s first formula for the logarithm of the gamma functio...