On the zeros of derivatives of Bessel functions

In this paper we are interested in the behaviour respect tov of thekth positive zeroc′ _vk of the derivative of the general Bessel functionC _v(x)=J _v(x)cosα?Y _v(x)sinα, 0≤α<π, whereJ _v(x) andY _v(x) indicate the Bessel functions of first and second kind respectively. It is well known that forc′ _vk>∥v∥,c′ _vk increases asv increases. Here we prove several additional properties forc′ _vk. Our main result is thatc′ _vk is concave as a function ofv, whenc′ _vk>∥v∥>0. This implies the concavity ofc′ _vk for everyk=2,3, ?. In the case of the zerosJ′ _vk of _d ^dx J _v(x) we extend this property tok=1 for everyv≥0.     

A convexity property of zeros of Bessel functions

Fork=1, 2,... letj _vk andc _vk be thek-th positive zeros of the Bessel function $$C_v \left( x \right) = C_v \left( {\alpha ;x} \right) = J_v \left( x \right)\cos \alpha - Y_v \left( x \right)\sin \alpha , 0 \leqslant \alpha< \pi$$ whereY _v (X) is the Bessel function of the second kind. Using the notationj _vκ =C _vk withκ=k?α/π introduced in [3] we show that the functionj _vκ +f(v) is convex with respect toυ≥0 forκ≥0.7070..., wheref(υ) is defined in the theorem of section 2. As an application we find the inequality 0 >j _0κ +j _1κ ? 2κπ > log 8/9, where κ≥0.7070....     

An approximation for the zeros of Bessel functions

Summary LetC _vk be thekth positive zero of the cylinder functionC _v(x)=cosJ _v(x)–sinY _v(x), whereJ _v(x),Y _v(x) are the Bessel functions of first kind and second kind, resp., andv>0, 0<. Definej _vk byj _vk=C _vk with . Using the notation 1/K=, we derive the first two terms of the asymptotic expansion ofj _vk in terms of the powers of at the expense of solving a transcendental equation. Numerical examples are given to show the accuracy of this approximation.Dedicated to the memory of Professor Lothar CollatzThis work has been supported by the Hungarian Scientific Grant No. 6032/6319     

Convergence of solutions of a second order Cauchy problem

Let j_νk, y_νk and c_νk denote the kth positive zeros of the Bessel functions J_ν(x), Y_ν(x) and of the general cylinder function C_ν(x), respectively. We show, among other things, that, for k = 2, 3,… and 0 < ν < ∞, c_νk is a concave function of ν, c_νk > ν + c_0k and $\text{c}{}_{\mathrm{\nu k}}\text{[v + (}\text{2}\text{π}\text{)c}{}_{\mathrm{0k}}\text{]}$ decreases as ν increases. In the cases of j_νk and y_νk, these results hold also for k = 1.     

Power-series unification and reversion

In analysis, it is sometimes necessary to unite a pair of power series into a single power series. If x = x(z) = Σ_ja_jz^j and y = y(z) = Σ_jb_jz^j are given power series, then by eliminating the common parameter z, the power-series unification is obtained: y = y(x) = Σ_kc_kx^k, where the coefficients c_k are to be determined in terms of the given power-series coefficients a_j and b_j. In a special case that y = z, the power-series reversion is obtained: z = z(x) = Σ_kd_kx^k, where the coefficients d_k are to be expressed in terms of the original power-series coefficients a_j. In this paper, explicit and recurrent formulas for the desired coefficients are derived. A simple technique of matrix formulation is developed for simplicity of computation. Finally, a complete computer program with a typical example is presented.     

An upper bound for the zeros of the derivative of Bessel functions

Letj _vk ^′ denotes thekth positive zero of the derivativeJ _v ^′ (x)=dJ _v (x)/dx of Bessel functionJ _v (x) fork=1, 2,…. We establish the upper bound

$$j'_{\nu k}< \nu + a_k \left( {\nu + \frac{{{\rm A}_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} + \frac{3}{{10}}a_k^2 \left( {\nu + \frac{{A_k^3 }}{{a_k^3 }}} \right)^{\frac{1}{3}} , \nu \geqslant 0, k = 1,2, \ldots $$     

An asymptotic relation for the zeros of Bessel functions

We define the function j_νκ for all real κ > 0 as follows: for κ = 1, 2, … the j_νκ denotes the kth positive zero of the Bessel function J_ν(z) of first kind and for k ? 1 < κ < k, j_νκ denotes the kth positive zero of the cylinder Bessel function C_ν(z) = cos αJ_ν(z) ? sin αY_ν(Z) with α = (k ? ν)π (see [2]), where Y_ν(x) is the Bessel function of second kind. We introduce the function ι(x) for x > ? 1,

$\text{l(x)=}\text{lim}\text{κ→∞}\text{j}{}_{\mathrm{\kappa ,x,\kappa }}\text{κ}$

. and we prove, among other things, the inequality $\text{j}{}_{\mathrm{\nu \kappa }}\text{< κι(}\text{ν}\text{κ}\text{)}$. Moreover, we find the first three terms of the asymptotic expansion of ι(x), for large values of x and other properties of this function.     

On the power of the zeros of Bessel functions

For ν≥0 let c_νk be the k-th positive zero of the cylinder functionC _v(t)=J _v(t)cosα-Y _v(t)sinα, 0≤α>π whereJ _ν(t) andY _ν(t) denote the Bessel functions of the first and the second kind, respectively. We prove thatC _v,k ^1+H(x) is convex as a function of ν, ifc _νk≥x>0 and ν≥0, whereH(x) is specified in Theorem 1.1.     

A monotonicity property of bessel functions

Some monotonicity results are given for the remainder terms in the asymptotic expansion for x → ∞ of the function J_v(x)J_v+n(x) + Y_v(x Y_v+n(x), v ε R, n ε Z.     

Characterization of a class of bivariate distribution functions

Let F_X,Y(x,y) be a bivariate distribution function and P_n(x), Q_m(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions F_X(x) and F_Y(y), respectively. Some necessary conditions are derived for the co-efficients c_n, n = 0, 1, 2,…, if the conditional expectation E[P_n(X) ∥ Y] = c_nQ_n(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.     

A general class of fuzzy operators,the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators

In this paper we examine operators which can be derived from the general solution of functional equations on associativity. We define the characteristics of those functions f(x) which are necessary for the production of operators. We shall show, that with the help of the negation operator for every such function f(x) a function g(x) can be given, from which a disjunctive operator can be derived, and for the three operators the DeMorgan identity is fulfilled. For the fulfillment of the DeMorgan identity the necessary and sufficient conditions are given.We shall also show that an f_λ(x) can be constructed for every f(x), so that for the derived k_λ(x,y) and d_λ(x,y) lim_λ→∞k_λ(x,y) and lim_λ→∞d_λ(x,y) = max(x,y).As Yager's operator is not reducible, for every λ there exists an α, for which, in case x < α and y<α, k_λ(x,y) = 0.We shall give an f(x) which has the characteristics of Yager's operator, and which is strictly monotone.Finally we shall show, that with the help of all those f(x), which are necessary when constructing a k(x,y), an F(x) can be constructed which has the properties of the measures of fuzziness introduced by A. De Luca and S. Termini. Some classical fuzziness measures are obtained as special cases of our system.     

Induced path transit function, monotone and Peano axioms

The induced path transit function J(u,v) in a graph consists of the set of all vertices lying on any induced path between the vertices u and v. A transit function J satisfies monotone axiom if x,y∈J(u,v) implies J(x,y)⊆J(u,v). A transit function J is said to satisfy the Peano axiom if, for any u,v,w∈V,x∈J(v,w), y∈J(u,x), there is a z∈J(u,v) such that y∈J(w,z). These two axioms are equivalent for the induced path transit function of a graph. Planar graphs for which the induced path transit function satisfies the monotone axiom are characterized by forbidden induced subgraphs.     

“Best possible” upper bounds for the first positive zeros of Bessel functions — the finite part

It is shown here that the first three terms of the asymptotic expansion of j_vk, k = 1, 2, 3, provide an upper bound for j_vk in 0 < v ⩽ 10, hence a “best possible” upper bound. Lang and Wong have shown that this is true also for 10 < v < ∞ when k = 1 and 2, so that these “best possible” upper bounds hold in the entire interval 0 < v < ∞ in these cases. We include supplementary comments on lower bounds in 0 ⩽ v ⩽ 10.     

Über komplementäre Ungleichungen mit drei Mittelwerten. Teil 1

Earlier investigations are extended to inequalities with three means of the formf(M _? (x;α),M _Ψ (y;α))?M _χ (f(x,y);α)≧0 (I). Replacing the given basic sets (x)=(x ₁,...,x _n ) and (y)=(y ₁,...,y _n ) by two suitably chosen sets (u)=(u ₁,...,u _m ) and (v)=(v ₁,...,v _m ), lower or upper bounds on the left side of (I) can be obtained. In the case of upper bounds these inequalities are complementary to (I). In general, the numberm is not less than 4; it may be reduced under additional hypotheses. Some examples (inequalities complementary to some additive inequalities) are given.     

Algorithm for the computation of Bessel function integrals

A fortran subroutine is given for the computation of integrals of the form ∫^c₀f(x)J_v(αx)dx, where v = 0, 1,…,10.     

A weighted geometric regularity from order restricted statistical inference

In Euclideank-space, the cone of vectors x = (x ₁,x ₂,...,x _k ) satisfyingx ₁ ≤x ₂ ≤ ... ≤x _k and $\sum\nolimits_{j = 1}^k {x_j } = 0$ is generated by the vectorsv _j = (j ?k, ...,j ?k,j, ...,j) havingj ?k’s in its firstj coordinates andj’s for the remainingk ?j coordinates, for 1 ≤j <k. In this equal weights case, the average angle between v _i and v _j over all pairs (i, j) with 1 ≤i <j <k is known to be 60°. This paper generalizes the problem by considering arbitrary weights with permutations.     

The induced path function, monotonicity and betweenness

The geodesic interval function I of a connected graph allows an axiomatic characterization involving axioms on the function only, without any reference to distance, as was shown by Nebeský [20]. Surprisingly, Nebeský [23] showed that, if no further restrictions are imposed, the induced path function J of a connected graph G does not allow such an axiomatic characterization. Here J(u,v) consists of the set of vertices lying on the induced paths between u and v. This function is a special instance of a transit function. In this paper we address the question what kind of restrictions could be imposed to obtain axiomatic characterizations of J. The function J satisfies betweenness if w∈J(u,v), with w≠u, implies u∉J(w,v) and x∈J(u,v) implies J(u,x)⊆J(u,v). It is monotone if x,y∈J(u,v) implies J(x,y)⊆J(u,v). In the case where we restrict ourselves to functions J that satisfy betweenness, or monotonicity, we are able to provide such axiomatic characterizations of J by transit axioms only. The graphs involved can all be characterized by forbidden subgraphs.     

An approximation property of exponential functions

We solve the inhomogeneous linear first order differential equations of the form y′(x) ? λy(x) = Σ _m=0 ^∞ a _m (x ? c) ^m , and prove an approximation property of exponential functions. More precisely, we prove the local Hyers-Ulam stability of linear first order differential equations of the form y′(x) = λy(x) in a special class of analytic functions.     

Singular (k,n − k) boundary value problems between conjugate and right focal

For 1 ⩽k⩽n − 1 and 0 ⩽q⩽k − 1, solutions are obtained for the boundary value problem, (−1)^n−k = f(x,y), y⁽ⁱ⁾=0, 0⩽i⩽k − 1, and y⁽ⁱ⁾ = 0, q⩽j⩽n − k + q − 1, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone.