advertisement

Volume 9 (2008), Issue 3, Article 83, 5 pp. ON EQUIVALENCE OF COEFFICIENT CONDITIONS. II L. LEINDLER B OLYAI I NSTITUTE U NIVERSITY OF S ZEGED A RADI VÉRTANÚK TERE 1 H-6720 S ZEGED , H UNGARY [email protected] Received 28 August, 2008; accepted 16 September, 2008 Communicated by H. Bor A BSTRACT. An additional theorem is proved pertaining to the equiconvergence of numerical series. Key words and phrases: Numerical series, equiconvergence. 2000 Mathematics Subject Classification. 26D15, 40D15. 1. I NTRODUCTION In the papers [2], [3] and [4] we have studied the relations of the following sums: ∞ X S1 := cqn µn , n=1 S2 := S3 := S4 := ∞ X λn ∞ X n=1 k=n ∞ X n X λn ! cqk cqk k=1 ∞ X νn+1 −1 n=1 S2∗ := , ! pq n=1 λn p q X k=νn S3∗ := , ∞ X λn µ−1 n n=1 k=1 ∞ X ∞ X λn µ−1 n n=1 ! pq cqk , S4∗ := n X ∞ X n=1 p ! q−p λk , p ! q−p λk , k=n λn λn µν n p q−p , where 0 < p < q, λ := {λn } and c := {cn } are sequences of nonnegative numbers, ν := {νm } is a subsequence of natural numbers, and µ := {µn } is a certain nondecreasing sequence of positive numbers. In [2] we verified that S2 < ∞ if and only if there exists a µ satisfying the conditions S1 < ∞ and S2∗ < ∞. Similarly S3 < ∞ if and only if S1 < ∞ and S3∗ < ∞. In [3] we showed that S4 < ∞ if and only if there exists a µ such that S1 < ∞ and S4∗ < ∞. 242-08 2 L. L EINDLER Recently, in [4], we proved that if µn := Λ(1) n Cnp−q , where ∞ X Cn := !1/q cqk and Λ(1) n := k=n then the sums S1 , S2 and S2∗ are already equiconvergent. Furthermore if !1/q n X p−q µn := Λ(2) where C̃n := cqk n C̃n , n X λk , k=1 and k=1 Λ(2) n := ∞ X λk , k=n then the sums S1 , S3 and S3∗ are equiconvergent. Comparing the results proved in [4] and that of [2] and [3], we can observe that in the former one the explicit sequences {µn } are determined, herewith they state more than the outcomes of [2] and [3], where only the existence of a sequence {µn } is proved. Furthermore, in [4] the equiconvergence of these concrete sums are guaranteed, too. However the equiconvergence in [4] is proved only in connection with the sums S2 and S3 , but not for S4 . This is a gap or shortcoming at these investigations. The aim of this note is closing this gap. Unfortunately we cannot give a complete solution, namely our result to be verified requires an additional assumption on the sequence λ. In particular, λ should be quasi geometrically increasing, that is, we assume that there exist a natural number N and K ≥ 1 such that λn+N ≥ 2λn and λn ≤ Kλn+1 hold for all n. Then we can give an explicit sequence µ such that the sums S1 , S4 and S4∗ are already equiconvergent. We also show that without some additional requirement on λ the equiconvergence does not hold. See the last part. Thus the following open problem can be raised: What is the weakest additional assumption on sequence λ which ensures the equiconvergence of these sums? 2. R ESULT Theorem 2.1. If 0 < p < q, c := {cn } is a sequence of nonnegative numbers, ν := {νm } is a subsequence of natural numbers, and λ := {λn } is a quasi geometrically increasing sequence, and for νm ≤ n < νm+1 ! pq −1 ∞ X µn := λm cqk , m = 0, 1, . . . , k=νm then the sums S1 , S4 and S4∗ are equiconvergent. 3. L EMMA In order to verify our theorem, first we shall prove a lemma regarding the equiconvergence of two special series. Lemma 3.1. Let 0 < α < 1, a := {an } be a sequence of nonnegative numbers, ν := {νm } be a subsequence of natural P numbers, and κ := {κm } be a quasi geometrically increasing sequence. Furthermore let Ak := ∞ n=k an , and for νm ≤ n < νm+1 let µn := κm Aα−1 νm , m = 0, 1, . . . . Then (3.1) σ1 := ∞ X an µ n < ∞ n=1 J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 83, 5 pp. http://jipam.vu.edu.au/ E QUIVALENCE OF C OEFFICIENT C ONDITIONS 3 holds if and only if ∞ X σ2 := (3.2) κm Aανm < ∞. m=1 Proof of Lemma 3.1. Before starting the proofs we note that the following inequality m X (3.3) κn ≤ K κm n=1 holds for all m, subsequent to the fact that κ is a quasi geometrically increasing sequence (see e.g. [1, Lemma 1]). Here and later on K denotes a constant that is independent of the parameters. Furthermore we verify a useful inequality. If 0 ≤ a < b, 0 < α < 1 and b α − aα = α ξ α−1 , (3.4) b−a then ξ ≥ α1/(1−α) b =: ξ0 , namely if a = 0 then ξ = ξ0 . Hence we get that α ξ α−1 ≤ bα−1 . (3.5) Now we show that (3.1) implies (3.2). Since An & 0, thus, by (3.3), ∞ ∞ ∞ X X X α κm Aνm = κm (Aανn − Aανn+1 ) m=1 = m=1 ∞ X n=m (Aανn − Aανn+1 ) n=1 ≤K (3.6) n X κm m=1 ∞ X κn (Aανn − Aανn+1 ). n=1 Using the relations (3.4) and (3.5) we obtain that ! νn+1 −1 X Aανn − Aανn+1 = ak α ξ α−1 ≤ νn+1 −1 X k=νn ! ak Aα−1 νn . k=νn This and (3.6) yield that ∞ X κm Aανm ≤K m=1 ∞ X νn+1 −1 κn Aα−1 νn n=1 X ak = K ∞ νn+1 X X−1 ak µ k . n=1 k=νn k=νn Herewith the implication (3.1) ⇒ (3.2) is proved. The proof of (3.2) ⇒ (3.1) is very easy. Namely ∞ X an µ n = n=ν1 ∞ νm+1 X X−1 an µ n m=1 n=νm = ≤ ∞ X m=1 ∞ X νm+1 −1 κm Aα−1 νm X an n=νm κm Aανm , m=1 J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 83, 5 pp. http://jipam.vu.edu.au/ 4 L. L EINDLER that is, (3.2) ⇒ (3.1) is verified. Thus the proof is complete. 4. P ROOF OF T HEOREM 2.1 p , q We shall use the result of Lemma 3.1 with α = P∞ q k=n ck and for νm ≤ n < νm+1 ∞ X µ n = µν m = λ m (4.1) an = cqn and κm = λm . Then An = ! p−q q cqk . k=νm Then σ1 = S1 , thus by Lemma 3.1, S1 < ∞ implies that σ2 < ∞, that is, ! pq ! pq νm+1 −1 ∞ ∞ ∞ X X X X cqn (4.2) S4 = λm ≤ λm cqn = σ2 . m=1 n=νm m=1 n=νm Moreover, by (4.1), S4∗ = ∞ X n=1 λn ∞ X cqk p q−p ! q−p q = k=νn ∞ X n=1 λn ∞ X ! pq cqk = σ2 , k=νn thus S1 < ∞ implies that both S4 < ∞ and S4∗ < ∞ hold. Conversely, if S4 < ∞, then it suffices to show that σ2 = S4∗ < ∞ also holds. Applying the inequality X α X ak ≤ aαk , 0 < α ≤ 1, ak ≥ 0, and (3.3), we obtain that σ2 = ∞ X λm Ap/q νm ≤ m=1 = νn+1 −1 n=1 k=νn ∞ X X ! pq cqk n X n=1 νn+1 −1 n=m k=νn X ! pq cqk λm m=1 ! pq νn+1 −1 λn ∞ X λm m=1 ∞ X ≤K ∞ X X cqk k=νn = KS4 < ∞. This, (4.2) and, by Lemma 3.1, the implication σ2 < ∞ ⇒ σ1 = S1 < ∞ complete the proof of Theorem 2.1. Proof of the necessity of some additional assumption on λ. Let p = 1, q = 2, λn = log n, νn = n and m−3 if n = 2m , cn := 0 otherwise. Then ∞ X log 2m S4 = < ∞, m3 m=2 J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 83, 5 pp. http://jipam.vu.edu.au/ E QUIVALENCE OF C OEFFICIENT C ONDITIONS 5 but S1 < ∞ and S4∗ < ∞ cannot be fulfilled simultaneously. Namely, then with a nondecreasing sequence {µn } the conditions ∞ X S1 = m−6 µ2m < ∞ m=1 and S4∗ = ∞ X log2 m <∞ µ m m=2 yield a trivial contradiction. R EFERENCES [1] L. LEINDLER, On equivalence of coefficient conditions with applications, Acta Sci. Math. (Szeged), 60 (1995), 495–514. [2] L. LEINDLER, On equivalence of coefficient conditions and application, Math. Inequal. Appl. (Zagreb), 1 (1998), 41–51. [3] L. LEINDLER, On a new equivalence of coefficient conditions and applications, Math. Inequal. Appl., 10(2) (1999), 195–202. [4] L. LEINDLER, On equivalence of coefficient conditions, J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 8. [ONLINE: http://jipam.vu.edu.au/article.php?sid=821]. J. Inequal. Pure and Appl. Math., 9(3) (2008), Art. 83, 5 pp. http://jipam.vu.edu.au/