Inclusion theorems associated with a certain new family of asymptotically and statistically equivalent functions
Authors
H. M. Srivastava
 University of Victoria, Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada \(\&\) Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China, Victoria, British Columbia V8W 3R4, Canada.
 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China.
Ekrem Savaş
 department of Mathematics, Istanbul Ticaret (Commerce) University, Sutluce (Beyglu), TR34672 UskudarIstanbul, Turkey.
Richard F. Patterson
 Department of Mathematics and Statistics, University of North Florida, Jacksonville, Florida 32224, U. S. A..
Abstract
The aim of this paper is to introduce and investigate some new
definitions which are interrelated to the notions of asymptotically \(
I_\lambda\)statistical equivalence of multiple \(L\) and strongly
\(I_\lambda\)asymptotic equivalence of multiple \(L\).
Indeed, instead of sequences, the
authors make use of two nonnegative realvalued Lebesgue measurable
functions in the open interval \((1,\infty)\) and present a series of
inclusion theorems associated with these new definitions. Furthermore, in
connection with one of the main results which are proven in this paper, a
closelyrelated \(open\) \(problem\) is posed for the interested reader.
Keywords
 Ideals and filters
 \(\mathcal{I}\)statistical convergence
 \(I_{\lambda}\)statistical convergence
 \(\lambda\)statistical convergence
 de la Vallée Poussin method
MSC
References

[1]
M. Aldhaifallah, K. S. Nisar, H. M. Srivastava, M. Mursaleen, Statistical \(\Lambda\)convergence in probabilistic normed spaces, J. Funct. Spaces, 2017 (2017), 7 pages.

[2]
N. L. Braha, V. Loku, H. M. Srivastava, \(\Lambda^2\)Weighted statistical convergence and Korovkin and Voronovskaya type theorems, Appl. Math. Comput., 266 (2015), 675–686.

[3]
H. Cakalli, A study on statistical convergence, Funct. Anal. Approx. Comput., 1 (2009), 19–24.

[4]
J. Connor, E. Savaş, Lacunary statistical and sliding window convergence for measurable functions, Acta Math. Hungar, 145 (2015), 416–432.

[5]
P. Das, P. Kostyrko, W. Wilczynski, P. Malik, I and I*convergence of double sequences, Math. Slovaca, 58 (2008), 605–620.

[6]
P. Das, E. Savaş, S. K. Ghosal, On generalizations of certain summability methods using ideals , Appl. Math. Lett., 24 (2011), 1509–1514.

[7]
K. Dems, On ICauchy sequences, Real Anal. Exchange, 30 (2004/2005), 123–128.

[8]
H. Fast, Sur la convergence ststistique, Colloq. Math., 2 (1951), 241–244.

[9]
A. R. Freedman, J. J. Sember, M. Raphael, Some Cesàro type summability spaces, Proc. London Math. Soc., 37 (1978), 508–520

[10]
J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313.

[11]
H. Gumuş, E. Savaş, On \(S^L_\lambda(I)\)asymptotically statistical equivalent sequences , Proceedings of the International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2012), 2012 (2012), 936–941.

[12]
U. Kadak, On weighted statistical convergence based on (p, q)integers and related approximation theorems for functions of two variables, J. Math. Anal. Appl., 443 (2016), 752–764.

[13]
U. Kadak, N. L. Braha, H. M. Srivastava, Statistical weighted Bsummability and its applications to approximation theorems, Appl. Math. Comput., 302 (2017), 80–96.

[14]
P. Kostyrko, W. Wilczynski, T. Salat, Iconvergence, Real Anal. Exchange, 26 (2000), 669–686.

[15]
V. Kumar, A. Sharma, On asymptotically generalized statistical equivalent sequences via ideal, Tamkang J. Math., 43 (2012), 469–478.

[16]
B. K. Lahiri, P. Das, I and I*convergence in topological spaces, Math. Bohem., 130 (2005), 153–160.

[17]
J. L. Li, Asymptotic equivalence of sequences and summability, Internat. J. Math. Math. Sci., 20 (1997), 749–758.

[18]
M. Marouf, Asymptotic equivalence and summability, Internat. J. Math. Sci., 16 (1993), 755–762.

[19]
M. Mursaleen, \(\lambda\)Statistical convergence, Math. Slovaca, 50 (2000), 111–115.

[20]
M. Mursaleen, A. Alotaibi, On \(\jmath\)convergence in random 2normed spaces, Math. Slovaca, 61 (2011), 933–940.

[21]
M. Mursaleen, V. Karakaya, M. Ertürk, F. Gürsoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput., 218 (2012), 9132–9137.

[22]
M. Mursaleen, S. A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Math. Slovaca, 62 (2012), 49–62.

[23]
M. Mursaleen, H. M. Srivastava, S. K. Sharma, Generalized statistically convergent sequences of fuzzy numbers, J. Intelligent Fuzzy Systems, 30 (2016), 1511–1518.

[24]
F. Nuray, \(\lambda\)Strongly summable and \(\lambda\)statistically convergent functions, Iranian J. Sci. Tech. Trans. A Sci., 34 (2010), 335–339.

[25]
R. F. Patterson, On asymptotically statistically equivalent sequences, Demostratio Math., 6 (2003), 149–153.

[26]
A. Sahiner, M. Gurdal, S. Saltan, H. Gunawan, Ideal convergence in 2normed spaces, Taiwanese J. Math., 11 (2007), 1477–1484.

[27]
T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980), 139–150.

[28]
E. Savaş, Asequence spaces in 2normed space defined by ideal convergence and an Orlicz function, Abstr. Appl. Anal., 2011 (2011), 9 pages.

[29]
E. Savaş, Generalized summability methods of functions using ideals, Proceedings of the International Conference on Advancements in Mathematical Sciences (Antalya, Turkey), 2015 (2015), 5 pages.

[30]
E. Savaş, On some new sequence spaces in 2normed spaces using ideal convergence and an Orlicz function, J. Inequal. Appl., 2010 (2010), 8 pages.

[31]
E. Savaş, On generalized statistical convergence in random 2normed space, Iran. J. Sci. Technol. Trans. A Sci., 36 (2012), 417–423.

[32]
E. Savaş, On \(\jmath\)asymptotically lacunary statistical equivalent sequences, Adv. Difference Equ., 2013 (2013), 7 pages.

[33]
E. Savaş, On generalized statistically convergent function via ideals, Appl. Math. Inform. Sci., 10 (2016), 943–947.

[34]
E. Savaş, \(\Delta^m\)Strongly summable sequences spaces in 2normed spaces defined by ideal convergence and an Orlicz function, Appl. Math. Comput., 217 (2010), 271–276.

[35]
R. Savaş, M. Basarir, (\(\sigma,\lambda\))Asymptotically statistical equivalent sequences, Filomat, 20 (2006), 35–42.

[36]
E. Savaş, P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett., 24 (2011), 826–830.

[37]
E. Savaş, H. Gumuş, A generalization on \(\jmath\)asymptotically lacunary statistical equivalent sequences, J. Inequal. Appl., 2013 (2013), 9 pages.

[38]
I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.

[39]
H. M. Srivastava, M. Et, Lacunary statistical convergence and strongly lacunary summable functions of order \(\alpha\), Filomat, 31 (2017), 1573–1582.