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Chaundry and Jolliffe [1] proved that if a k is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence of the series ∞ k=1 a k sin kx is lim k→∞ ka k = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu [4] generalized this classical result. In this paper we propose new classes of sequences for which we get the extended version of their results. Moreover, we generalize the results of S. Tikhonov [2] on the L 1 -convergence of Fourier series.

Abstract. Let a single sine series ( * ) P ∞ k=1 a k sin kx be given with nonnegative coefficients {a k }. If {a k } is a "mean value bounded variation sequence" (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series ( * ) is that ka k → 0 as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series ( * * ) P ∞ k=1 P ∞ l=1 c kl sin kx sin ly, even with complex coefficients {c kl }. We also give a uniform boundedness test for the rectangular partial sums of series ( * * ), and slightly improve the results on single sine series.

The Hermite-Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95-106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.

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