J. Numer. Anal. Approx. Theory, vol. 43 (2016) no. 2, pp. 177–182 ictp.acad.ro/jnaat
ON α-CONVEX SEQUENCES OF HIGHER ORDER∗
XHEVAT Z. KRASNIQI†
Abstract. Many important applications of the class of convex sequences came across in several branches of mathematics as well as their generalizations. In this paper, we have introduced a new class of convex sequences, the class ofα- convex sequences of higher order. In addition, the characterizations of sequences belonging to this class have been shown.
MSC 2010. 26A51, 26A48, 26D15.
Keywords. Sequence, convexity,α-convexity, (p, q;r)-convexity, starshaped se- quence, higher order convexity.
1. INTRODUCTION
The class of convex sequences is one of most important subclass of the class of real sequences. This class is raised as a result of some efforts to solve several problems in mathematics. Naturally, the sequences that belong to that class, have useful applications in some branches of mathematics, in particular in mathematical analysis. For instance, such sequences are widely used in theory of inequalities (see [13], [7], [8]), in absolute summability of infinite series (see [1], [2]), and in theory of Fourier series, related to their uniform convergence and the integrability of their sum functions (see as example [6], page 587). Here, in this paper, we are going to introduce a new particular class of convex sequences, which indeed generalizes an another class of convex sequences introduced previously by others. In order to do this, we need first to recall some notations and notions as follows in the sequel.
Let (an)∞n=0 be a real sequence. It is previously defined that
40an=an, 41an=an+1−an, 4man=4(4m−1an), m, n= 0,1, . . . , and throughout the paper we shall write 4an instead of41an.
The following definition presents the concept of convexity of higher order.
Definition 1. A sequence(an)∞n=0 is said to be convex of higher order (or m-convex) if
4man≥0,
†Department of Mathematics and Informatics, Faculty of Education, University of Pr- ishtina “Hasan Prishtina”, Avenue “Mother Theresa” 5, 10000 Prishtina, Kosovo, e-mail:
for all n≥0.
Next Lemma has been proved by Gh. Toader, which characterizes the convex sequences of higher order (or m-convex).
Lemma 2. [11] The sequence (an)∞n=0 is convex of order m (m ∈N, fixed) iff
an=
n
X
k=0
n+m−k−1 m−1
bk
with bk≥0, for k≥m.
Various generalizations of convexity were studied by many authors. For instance,p-convexity (see [5]), (p, q)-convexity (see [4]), and (p, q;r)-convexity [3].
Two other classes of sequences, the so-called, starshaped sequences and α- convex sequences have been introduced in [9] and [10].
Indeed, let throughout this paper beα∈[0,1].
Definition 3. A sequence(an)∞n=0 is called α-convex if the sequence α(an+1−an) + (1−α)an−an 0∞n=1
is increasing.
Definition 4. A sequence(an)∞n=0 is called starshaped if an+1−a0
n+ 1 ≥ an−a0 n for n≥1.
The class of starshaped sequences of higher order has been introduced in [12]:
Definition 5. A sequence(an)∞n=0 is called starshaped of order m if 4m−1
an+1−a0 n+ 1
≥0 for all n≥1 and m∈ {2,3, . . .}.
Next Lemma characterizes the starshaped sequences of order m.
Lemma 6. [12] The sequence (an)∞n=0 is starshaped sequences of order m (m∈N, fixed) iff
an=n
n
X
k=1
n+m−k−2 m−2
ck+c0
with ck≥0, for k > m.
Here, we introduce a new class of sequences as follows:
Definition 7. A sequence (an)∞n=0 is called (m, α)-convex (orα-convex of order m) if the sequence
α4m−1(an+1−an) + (1−α)4m−1 an−an 0∞
n=1
is increasing for all n∈ {0,1, . . .}, and for arbitrary fixed m, m∈N.
Remark 8. We note that: (1, α)-convexity is the same with α-convexity, (m,1)-convexity is the same with m-convexity, (m,0)-convexity is the same with star-shapedness of order m, (1,1)-convexity is the same with ordinary convexity, and (1,0)-convexity is the same with star-shapedness.
Characterizing (m, α)-convex sequences, we are going to accomplish the main aim of this paper.
2. MAIN RESULTS
First, we begin with:
Theorem 9. The sequence (an)∞n=0 is (m, α)-convex if and only if α4m+1(an) + (1−α)4m an−an 0≥0,
for all n∈ {0,1, . . .}, and for arbitrary fixed m, m∈N.
Proof. The proof of this statement is an immediate result of the Definition 7.
The proof is completed.
Form= 1 we obtain:
Corollary 10. [10] The sequence (an)∞n=0 isα-convex if and only if α42(an) + (1−α)an+1n+1−a0 − an−an 0≥0,
for all n∈ {0,1, . . .}.
Theorem 11. The sequence (an)∞n=0 is (m, α)-convex if and only if (an−a0+α[n(an+1−an)−(an−a0)])∞n=1,
is a starshaped sequence of order m, for arbitrary fixed m andm∈N. Proof. For the sake of brevity we denote
An:=an−a0+α[n(an+1−an)−(an−a0)], n∈ {1,2, . . .}, which can be rewritten as
An=αn(an+1−an) + (1−α)(an−a0), n∈ {1,2, . . .}.
Form= 1 we have (see also [10]) that (A0 = 0)
4Ann≥0⇐⇒α42(an) + (1−α)4 an−an 0≥0, for all n∈ {0,1, . . .}.
According to this, and since the operator 4is a linear one, then we have:
4mAnn≥0 ⇐⇒ 4(4(. . .(4
| {z }
m−times
(An/n))))≥0
⇐⇒ 4(4(. . .(4
| {z }
(m−1)−times
α42(an) + (1−α)4 an−an 0)))≥0
⇐⇒ α4m−1+2(an) + (1−α)4m−1+1 an−an 0≥0
⇐⇒ α4m+1(an) + (1−α)4m an−an 0≥0, for all n∈ {0,1, . . .}and for arbitrary fixedm,m∈N.
The proof is completed.
Form= 1 we obtain:
Corollary 12. [10] The sequence (an)∞n=0 isα-convex if and only if (an−a0+α[n(an+1−an)−(an−a0)])∞n=1,
is a starshaped sequence.
Theorem 13. The sequence (an)∞n=0 is(m, α)-convex if and only if it may be represented by
(1) an=n
n
X
k=1 ck
k −(n−1)c0, with
(2) 4m−1cn+2−n+1n cn+1
≥1−α14m−1cn+1n+1,
4m−1cn+2−n+1n cn+1≥0,4m−1cn+1n+1≥0, n≥2, and for arbitrary fixed m, m∈N.
Proof. On one hand, taking into account (1), we easy obtain 42(an) =cn+2−n+1n cn+1
and, consequently
4m+1(an) =4m−1cn+2−n+1n cn+1. (3)
On the other hand, using (1) again, we also have 4 an−an 0= cn+1n+1 and, thus
4m an−an 0=4m−1cn+1n+1. (4)
From (3) and (4) we obtain
α4m+1(an) + (1−α)4m an−an 0=
=α4m−1cn+2−n+1n cn+1+ (1−α)4m−1cn+1n+1. Subsequently, it follows that
α4m+1(an) + (1−α)4m an−an 0≥0 if and only if
4m−1cn+2− n+1n cn+1
≥1−α14m−1cn+1n+1.
The proof is completed.
Remark 14. Note that representation (1) has been presented for the first
time in [9].
Corollary 15 ([10]). The sequence (an)∞n=0 is α-convex if and only if it may be represented by
an=n
n
X
k=1 ck
k −(n−1)c0, with
cn+2 ≥1−α(n+1)1 cn+1, and cn≥0, n≥2.
Theorem 16. Let m∈Nbe fixed and α∈[0,1]. If the sequence (an)∞n=0 is (m, α)-convex, then it is (m, β)-convex.
Proof. The proof follows from Theorem 13. Indeed, let the sequence (an)∞n=0 be (m, α)-convex. Then, it may be represented by (1) with (2),
4m−1cn+2− n+1n cn+1≥1−α14m−1cn+1n+1, and
4m−1cn+2−n+1n cn+1≥0, 4m−1cn+1n+1≥0, n≥2.
Although, since 0≤β ≤α, we also have
α4m−1cn+2−n+1n cn+1≥1−β14m−1cn+1n+1,
with same conditions as above, which shows that the sequence (an)∞n=0 is (m, β)-convex as well.
The proof is completed.
Form= 1, as a particular case, we obtain
Corollary 17 ([10]). If the sequence (an)∞n=0 is α-convex, then it is β- convex, for 0≤β ≤α.
Acknowledgement. The author would like to thank the anonymous ref- eree for her/his remarks which averted some inaccuracies.
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Received by the editors: August 18, 2016.
