In this paper, we investigate the uniqueness problem of P(f)L(g) and P(g)L(f) when they share $$\alpha (z)$$ counting multiplicities, where L(h) represents any one of $$h^{(k)}(z),\; h(z+c),\; h(z+c)-h(z)$$ and $$h^{(k)}(z+c)$$ where k is a positive integer, c is a nonzero constant and $$\alpha (z)$$ is a nonzero small function with respect to both f(z) and g(z). The results of the paper generalize the recent results of Gao and Liu (Bull Korean Math Soc 59:155–166, 2022).

In the unorganized portion of his second notebook [6], Ramanujan has recorded modular equations corresponding to signatures 3 and 4. The proofs of the same can be found in Chapter 33 of [3]. In this article, we use the method of parametrization to establish a few new modular equations in signature 3 and 4 which are analogous to modular equations in signature 2 due to Ramanujan. Using the same method, we also prove a few modular equations in signatures 3 and 4, recorded by Ramanujan in his second notebook [6].

In this paper, we introduce weakly generalized (ψ, φ)-weak quasi contraction for four self-maps and establish a common fixed point theorem using weak compatible property.

Abstract In this paper we deal with Nil geometry, which is one of the homogeneous Thurston 3-geometries. We define the “surface of a geodesic triangle” using generalized Apollonius surfaces. Moreover, we show that the “lines” on the surface of a geodesic triangle can be defined by the famous Menelaus’ condition and prove that Ceva’s theorem for geodesic trianglesistruein Nil space. In our work we will use the projective model of Nil geometry described by E. Molnár in [6].

Abstract The purpose of the present paper is to find a necessary and sufficient condition for Miller-Ross-type Poisson distribution series to be in the class 𝒲δ(α, γ, β) of analytic functions with negative coefficients .Also, we investigate several inclusion properties of the classes 𝒮∗, 𝒦 and ℝτ(A, B) associated of the operator 𝕀ν,c m defined by this distribution. Further, we consider an integral operator related to Miller-Ross-type Poisson distribution series. Several corollaries and consequences of the main results are also considered.

Abstract The purpose of the present paper is to introduce and study new subclasses of analytic functions which generalize the classes of Janowski functions with q-derivative. We also study certain a convolution conditions, and apply the convolution conditions to get sufficient condition and the neighborhood results related to the functions in the class 𝒮q(A, B, α).

Abstract A permutation p of [k] = {1, 2, 3, …, k} is called Layman permutation iff i + p(i) is a Fibonacci number for 1 ≤ i ≤ k. This concept is introduced by Layman in the A097082 entry of the Encyclopedia of Integers Sequences, that is the number of Layman permutations of [n]. In this paper, we will study Layman permutations. We introduce the notion of the Fibonacci complement of a natural number, that plays a crucial role in our investigation. Using this notion we prove some results on the number of Layman permutations, related to a conjecture of Layman that is implicit in the A097083 entry of OEIS.

Abstract In this paper, semisimple semigroups, duo semigroups, right (left) zero semigroups, right (left) simple semigroups, semilattice of left (right) simple semigroups, semilattice of left (right) groups and semilattice of groups are characterized in terms of soft intersection semigroups, soft intersection ideals of semigroups. Moreover, soft normal semigroups are defined and some characterizations of semigroups with soft normality are given.

Abstract This paper is concerned with the analysis of an infinite-capacity batch arrival multi-server queueing system with Bernoulli feedback, synchronous multiple and single working vacation policies, waiting servers, reneging and retention of reneged customers. The steady-state solution of the queueing system is obtained by using probability generating function (PGF). In addition, important performance measures of the queueing system are derived. Then, a cost model is formulated in order to carry out the parameter optimization using genetic algorithm (GA). Finally, numerical study is presented in which various system performance measures are evaluated based on supposed numerical values given to the system parameters.

Abstract We show that certain known or new inequalities for the logarithm of circular hyperbolic functions imply bounds for exp(±x2) proved in [1].