On -convex sequences of higher order
August 18, 2016.
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 introduce a new class of convex sequences, the class of
MSC. 26A51, 26A48, 26D15.
Keywords. Sequence, 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
and throughout the paper we shall write
The following definition presents the concept of convexity of higher order.
A sequence
for all
Various generalizations of convexity were studied by many authors. For instance,
Two other classes of sequences, the so-called, starshaped sequences and
Indeed, let throughout this paper be
A sequence
is increasing.
A sequence
for
A sequence
for all
Here, we introduce a new class of sequences as follows:
A sequence
is increasing for all
We note that:
Characterizing
2 Main Results
First, we begin with:
The sequence
for all
The proof is completed.
For
The sequence
is a starshaped sequence of order
which can be rewritten as
For
for all
According to this, and since the operator
for all
The proof is completed.
For
The sequence
with
and, consequently
On the other hand, using (1) again, we also have
and, thus
Subsequently, it follows that
if and only if
The proof is completed.
The sequence
with
and
Let
and
Although, since
with same conditions as above, which shows that the sequence
The proof is completed.
If the sequence
The author would like to thank the anonymous referee for her/his remarks which averted some inaccuracies.
Bibliography
- 1
H. Bor, A new application of convex sequences, J. Class. Anal., 1 (2012), no. 1, pp. 31–34.
- 2
- 3
Xh. Z. Krasniqi, Some properties of
-convex sequences, Appl. Math. E-Notes, 15 (2015), pp. 38–45.- 4
- 5
I. B. Lacković, M. R. Jovanović, On a class of real sequences which satisfy a difference inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., No. 678–715 (1980), pp. 99–104 (1981).
- 6
B. Makarov, A. Podkorytov, Real Analysis: Measures, Integrals and Applications, Springer-Verlag London, 2013.
- 7
J. E. Pec̆arić, On some inequalities for convex sequences, Publ. Inst. Math. (Beograd) (N.S.), 33 (47) (1983), pp. 173–178.
- 8
- 9
- 10
Gh. Toader,
-convex sequences, Itinerant seminar on functional equations, approximation and convexity (Cluj-Napoca, 1983), pp. 167–168, Preprint, 83-2, Univ. "Babeş-Bolyai”, Cluj-Napoca, 1983.- 11
Gh. Toader, On the convexity of high order of sequences, Publ. Inst. Math. (Beograd) (N.S.), 43 (57) (1988), pp. 35–40.
- 12
Gh. Toader, Starshapedness and superadditivity of high order of sequences. Itinerant seminar on functional equations, approximation and convexity (Cluj-Napoca, 1985), pp. 227–234, Preprint, 85–6, Univ. "Babeş-Bolyai", Cluj-Napoca, 1985.
- 13