Serdica Mathematical Journal
Volume 47, Number 3, 2021
C O N T E N T S
·
Reddy, P. A., M. G. Rao, V. S. Rama Prasad.
On the lcm-sum function over arbitrary sets of integers
(pp. 179−190)
·
Waghamore, H. P., P. N. Raj.
Uniqueness results on meromorphic functions concerning their shift and differential polynomial
(pp. 191−212)
·
Rathod, A.
On algebroid functions that share one finite value with their derivative on annuli
(pp. 213−240)
·
Singh, G., G. Singh.
A note on subclasses of multivalent functions
(pp. 241−254)
·
Todorova, T. L.
On the lcm-sum function over arbitrary sets of integers
(pp. 255−272)
A B S T R A C T S
ON THE LCM-SUM FUNCTION OVER ARBITRARY SETS OF INTEGERS
P. Anantha Reddy
ananth_palle@yahoo.co.in
M. Ganeshwar Rao
ganehwararao_maths@cbit.ac.in
V. Siva Rama Prasad
vangalasrp@yahoo.co.in
2020 Mathematics Subject Classification:
Primary 11A25; Secondary 11N37.
Key words:
Zeta-function of S, unitary divisor, r-free integer, semi-r-free integer, unitary r-free integer, (k,r)-integer.
Let ℕ denote the set of all positive integers. For j, n ∈ ℕ, let (j, n) and [j, n] respectively denote their gcd and lcm. If S ⊆ ℕ and α is a real number then define LS, α(n) to be the sum of [j, n]α, where j ∈ {1, 2, 3, ..., n} for which (j, n) ∈ S. In this paper we obtain asymptotic formulae for the summatory functions of LS, a(n) and LS, −a(n), where a ∈ ℕ and a ≥ 2. Apart from deducing some results proved earlier for S = ℕ by Ikeda and Matsuoka, certain new asymptotic formulae are obtained here.
UNIQUENESS RESULTS ON MEROMORPHIC FUNCTIONS CONCERNING THEIR SHIFT AND DIFFERENTIAL POLYNOMIAL
Harina P. Waghamore
harina@bub.ernet.in,
harinapw@gmail.com
Preetham N. Raj
preethamnraj@bub.ernet.in,
preethamnraj@gmail.com
2020 Mathematics Subject Classification:
Primary 30D35.
Key words:
Meromorphic function, shift, differential polynomial, unicity, weighted sharing.
In this paper, we investigate the uniqueness of meromorphic functions by considering their shift, q-difference and differential polynomial. We obtain some results which extend and generalize the results given by Chao Meng and Gang Liu [On unicity of meromorphic functions concerning the shifts and derivatives. J. Math. Inequal. 14, no. 4 (2020), 1095−1112].
ON ALGEBROID FUNCTIONS THAT SHARE ONE FINITE VALUE WITH THEIR DERIVATIVE ON ANNULI
Ashok Rathod
ashokmrmaths@gmail.com
2020 Mathematics Subject Classification:
30D35.
Key words:
Value Distribution Theory, algebroid functions, share DM, annuli.
In this paper, we discuss the algebroid functions W(z) and W'(z) that share the value 1 CM (counting multiplicities) and share one finite value DM (different multiplicities) with derivative on annuli.
A NOTE ON SUBCLASSES OF MULTIVALENT FUNCTIONS
Gagandeep Singh
kamboj.gagandeep@yahoo.in
Gurcharanjit Singh
dhillongs82@yahoo.com
2020 Mathematics Subject Classification:
30C45, 30C50.
Key words:
Analytic functions, multivalent functions, subordination, Sãlãgean operator, coefficient estimates, distortion theorem, argument theorem.
In this paper, certain subclasses of strongly multivalent functions are introduced with generalized Sãlãgean operator. We establish various properties for these classes and the results obtained here will generalize some other known results.
DIOPHANTINE APPROXIMATION BY PRIME NUMBERS OF A SPECIAL FORM
Tatiana L. Todorova
tlt@fmi.uni-sofia.bg
2020 Mathematics Subject Classification:
11D75, 11N36, 11P32.
Key words:
Rosser's weights, vector sieve, circle method, almost primes, diophantine inequality.
We show that whenever δ > 0, η are reals and constants λi subject
to certain assumptions, there are infinitely many prime triples
p1, p2, p3 satisfying the inequality
|λ1p1 + λ2p2 + λ3p3 + η| < (max pj)−1/18 + δ
and such that, for each i ∈ {1, 2, 3}, pi + 2 has at most 7 prime factors. The proof uses Davenport−Heilbronn adaption of the circle method together with a vector sieve method.
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