Let be the n th , with , mean of the sequence . Then, by (2), we have
First, applying Abel’s transformation and then using Lemma 1, we get that
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
Now, when , applying Hölder’s inequality with indices k and , where , we get that
by virtue of the hypotheses of the theorem and Lemma 2. Finally, we have that
by virtue of the hypotheses of the theorem and Lemma 2. This completes the proof of the theorem. If we take and (resp. , and ), then we get a new result dealing with (resp. ) summability factors of infinite series. Also, if we take and , then we get another new result concerning the summability factors of infinite series. Furthermore, if we take as an almost increasing sequence, then we get the result of Bor and Seyhan under weaker conditions (see ). Finally, if we take , then we obtain Theorem B.