1. Introduction

Let \(G\) be a set. A function \(K\colon G\times G\to\mathbb{C}\) is called a positive definite kernel on \(G\times G\) if, for any natural number \(n\in\mathbb{N}\), for any collections \(\{x_k\}_{k=1}^n \subset G\) of points and \(\{c_k\}_{k=1}^n \subset \mathbb{C}\) of numbers, the following inequality holds:

$$ \sum_{k,p=1}^n c_k\overline c_pK(x_k,x_p)\ge 0.$$
(1.1)

The set of all such kernels will be denoted by the symbol \(\Phi(G\times G)\). A kernel \(K\in\Phi(G\times G)\) is said to be strictly positive definite if inequality (1.1) is strict under the condition that all the points in \(\{x_k\}_{k=1}^n\) are pairwise distinct (i.e., \(x_k\ne x_p\) for \(k\ne p\)) and the numbers in \(\{c_k\}_{k=1}^n\) are not all zero (i.e., \(|c_1|+\cdots+|c_n|>0\)).

Let \(G\) be a group (not necessarily Abelian) with group operation \(+\). A function \(f\colon G\to\mathbb{C}\) is said to be positive definite on \(G\) if the function \(K(x,y):=f(x-y)\) is a positive definite kernel on \(G\times G\). The set of all such functions will be denoted by the symbol \(\Phi(G)\).

For \(n=2\), \(x_1=x\), and \(x_2=y\), inequality (1.1) for \(K\in\Phi(G\times G)\) is of the form

$$ a_{1,1}|c_1|^2+a_{1,2}c_1\overline c_2+a_{2,1}c_2\overline c_1+ a_{2,2}|c_2|^2\ge 0,\qquad c_1,c_2\in\mathbb{C},$$
(1.2)

where \(a_{1,1}=K(x,x)\), \(a_{1,2}=K(x,y)\), \(a_{2,1}=K(y,x)\), and \(a_{2,2}=K(y,y)\). In (1.2), we take \(c_1=1\) and \(c_2=0\). Then \(a_{1,1}\ge 0\). Similarly, \(a_{2,2}\ge 0\). Therefore,

$$a_{1,2}c_1\overline c_2+a_{2,1}c_2\overline c_1\in\mathbb{R}\qquad \text{for any}\quad c_1,c_2\in\mathbb{C}.$$

In particular, for \(c_1\overline c_2=1\) and \(c_1\overline c_2=i\), we obtain \(2a:=a_{1,2}+a_{2,1}\in\mathbb{R}\) and \(2b:=(a_{1,2}-a_{2,1})i\in\mathbb{R}\), respectively. Therefore,

$$a_{1,2}=a-bi\qquad\text{and}\qquad a_{2,1}=a+bi=\overline a_{1,2}.$$

Let \(a_{1,2}=|a_{1,2}|e^{i\phi}\), where \(\phi\in\mathbb{R}\). As proved above, \(a_{2,1}=|a_{1,2}|e^{-i\phi}\). In (1.2), we take \(c_1=t\in\mathbb{R}\) and \(c_2=e^{i\phi}\). Then, for any \(t\in\mathbb{R}\), the following inequality holds:

$$a_{1,1}t^2+2|a_{1,2}|t+a_{2,2}\ge 0.$$

Therefore, \(|a_{1,2}|^2\le a_{1,1}a_{2,2}\).

Thus, inequality (1.2) immediately implies the following simple and well-known properties of positive definite kernels: if \(K\in\Phi(G\times G)\), then

$$K(x,x)\ge 0,\quad K(y,x)=\overline{K(x,y)},\quad |K(x,y)|^2\le K(x,x) K(y,y)\quad\textit{for all}\quad x,y\in G.$$

In the theory of positive definite kernels and functions, an important role is played by the following Krein inequality (see, e.g., [1, Chap. IV, Sec. 1]): if \(K\in\Phi(G\times G)\), then, for all \(x_1,x_2,x\in G\), the following inequality holds:

$$ |K(x_1,x)-K(x_2,x)|^2 \le K(x,x)\operatorname{Re} (K(x_1,x_1)+K(x_2,x_2)-2K(x_1,x_2)).$$
(1.3)

For functions \(f\in\Phi(G)\), inequality (1.3) first appeared in 1940 in Krein’s paper [2]. Somewhat earlier, for functions \(f\in\Phi(\mathbb{R})\), Krein’s student Artemenko proved that the continuity of \(\operatorname{Re}f\) at zero implies the uniform continuity of \(f\) on \(\mathbb{R}\). This result also follows from inequality (1.3) (this was noted implicitly in Krein’s paper [3]). In 2015, using Bochner’s theorem and the maximum principle for subharmonic functions, Gorin [4] generalized Krein’s inequality (1.3) to functions in \(\Phi(G)\) (see also his papers [5], [6]). For \(G=\mathbb{R}\), see also the paper [7] of Pevnyi and Sitnik.

The aim of the present paper is to prove new general inequalities of Krein–Gorin type for functions from \(\Phi(G\times G)\). The key tool in the paper is the well-known main inequality for such kernels, namely, the Cauchy–Bunyakovskii inequality for the special inner product generated by a given positive definite kernel.

In Sec. 2 of the present paper, we present a simple proof of the well-known inequality (2.2), which is, in our opinion, the main inequality for positive definite kernels. In addition, we give a criterion for the main inequality to become an equality. Under an appropriate choice of the system of points and the coefficients in the main inequality, we obtain new inequalities (see Corollary 8) and the well-known inequalities of Krein and Weil (2.14), and Gorin (2.19). It is shown that Ingham’s inequality (2.12) (and, in particular, Hilbert’s inequality) is, essentially, the main inequality for the positive definite function \(\sin(\pi x)/x\) on \(\mathbb{R}\) and for a system of integer points (see Corollary 5 and Remark 5). In Secs. 3 and 4, the main inequality is used to prove new inequalities of the type of the Krein–Gorin inequalities and Ingham’s inequality.

2. The main inequality and its corollaries

Let \(\mathfrak{L}\) be a linear space over the field \(\mathbb{C}\) of complex numbers. A function \((h,g)\colon\mathfrak{L}\times \mathfrak{L}\to\mathbb{C}\) is called a semi-inner product on \(\mathfrak{L}\) if, for all \(h,g,u\in \mathfrak{L}\) and \(\lambda,\mu\in\mathbb{C}\), the following conditions hold:

  1. (1)

    \((\lambda h+\mu u,g)=\lambda(h,g)+\mu(u,g)\);

  2. (2)

    \((h,g)=\overline{(g,h)}\);

  3. (3)

    \((h,h)\ge 0\).

If, additionally, \((h,h)>0\) for \(h\in \mathfrak{L}\setminus\{0\}\), then this function is called an inner product on \(\mathfrak{L}\). It follows from the properties of a semi-inner product that, for all \(h,g\in \mathfrak{L}\) and \(c_1,c_2\in\mathbb{C}\), the following inequality holds:

$$(c_1h+c_2g,c_1h+c_2g)=(h,h)|c_1|^2+(h,g)c_1\overline c_2+ (g,h)c_2\overline c_1+(g,g)|c_2|^2\ge 0.$$

It follows that if \((g,g)=0\), then \((h,g)=0\) for all \(h\in G\) (we must take \(c_1= t\in\mathbb{R}\) and \(c_2=(h,g)\)). From the relation

$$(u,u)=(g,g)((h,h)(g,g)-|(h,g)|^2)\ge 0,\qquad\text{where}\quad u:=(g,g)h-(h,g)g,$$

we obtain the Cauchy–Bunyakovskii inequality for semi-inner product:

$$ |(h,g)|^2\le (h,h)(g,g),\qquad h,g\in \mathfrak{L}.$$
(2.1)

In addition, the equality in (2.1) holds if and only if \((u,u)=0\), where \(u=(g,g)h-(h,g)g\). If the function \((\,\cdot\,{,}\,\cdot\,)\) is an inner product on \(\mathfrak{L}\), then the equality in (2.1) holds if and only if \((g,g)h-(h,g)g=0\).

The proof of the well-known inequality (2.2) (for functions \(f\in\Phi(G)\), see, e.g., Sasvári’s monographs [8, Theorem 1.4.1 (v)] and [9, Theorem 1.4.12 (v)]) reduces to the Cauchy–Bunyakovskii inequality (2.1) for the special inner product generated by a given positive definite kernel \(K\). In the proof of Theorem 1, we use the same linear space \(\mathfrak{H}\) and inner product (2.5) that are used in the proof of Aronszajn’s theorem on the existence of a Hilbert space with reproducing kernel \(K\) (see, e.g., the monographs [1, Chap. IV, Sec. 1] and [10, Part I, Sec. 2]).

Theorem 1 main inequality.

Let \(K\in\Phi(G\times G)\). Then, for all natural numbers \(n,m\in\mathbb{N}\), for arbitrary collections \(\{x_k\}_{k=1}^n, \{y_p\}_{p=1}^m \subset G\), for \(x\in G\), and for \(\{a_k\}_{k=1}^n, \{b_p\}_{p=1}^m \subset \mathbb{C}\), the following inequalities hold :

$$\begin{aligned} \, \biggl|\sum_{k=1}^n\,\sum_{p=1}^m K(x_k,y_p) a_k\overline b_p\biggr|^2&\le \sum_{k,p=1}^n K(x_k,x_p)a_k\overline a_p \sum_{k,p=1}^m K(y_k,y_p)b_k\overline b_p, \end{aligned}$$
(2.2)
$$\begin{aligned} \, \biggl|\sum_{k=1}^n a_k K(x_k,x)\biggr|^2&\le K(x,x)\sum_{k,p=1}^n K(x_k,x_p)a_k\overline a_p. \end{aligned}$$
(2.3)

Inequality (2.2) becomes an equality for certain natural numbers \(n,m\in\mathbb{N}\) and collections \(\{x_k\}_{k=1}^n, \{y_p\}_{p=1}^m \subset G\), and \(\{a_k\}_{k=1}^n, \{b_p\}_{p=1}^m \subset \mathbb{C}\) if and only if, for all \(t\in G\), the following equality holds :

$$\sum_{k,p=1}^m K(y_k,y_p)b_k\overline b_p \sum_{k=1}^{n}a_k K(x_k,t)= \sum_{k=1}^n\,\sum_{p=1}^m K(x_k,y_p)a_k\overline b_p \sum_{p=1}^{m}b_p K(y_p,t),\qquad t\in G.$$

In particular, the inequality \(|K(x,y)|^2\le K(x,x)K(y,y)\) becomes an equality for some \(x,y\in G\) if and only if, for all \(t\in G\), the following equality holds :

$$K(y,y)K(x,t)=K(x,y)K(y,t).$$

Remark 1.

It follows from Theorem 1 that if \(G\) is a group and \(f\in\Phi(G)\), then the inequality \(|f(x)|\le f(0)\) becomes an equality for some \(x\in G\) if and only if, for all \(t\in G\), the equality \(f(0)f(x+t)=f(x)f(t)\) holds. In particular, if \(f\in\Phi(G)\) and \(f(x)=\varepsilon f(0)\) for some \(x\in G\) and \(\varepsilon\in\mathbb{C}\), \(|\varepsilon|=1\), then, for all \(t\in G\) and \(n\in\mathbb{Z}\), the equality \(f(nx+t)=\varepsilon^n f(t)\) holds (see, e.g., [6, Lemma 1], [7, Corollary 1]).

Proof.

Inequality (2.3) follows from inequality (2.2) for \(m=1\), \(y_1=x\), and \(b_1=1\). Let us prove inequality (2.2).

Let \(K\in\Phi(G\times G)\), and let \(\mathfrak{H}\) be the linear span of the family

$$\{k_{\lambda}(t):=K(\lambda,t),\,t\in G\}_{\lambda\in G}$$

of functions. Obviously, \(\mathfrak{H}\) is a linear space over the field \(\mathbb{C}\) with zero element \(\phi(t)\equiv 0\) on \(G\). We define a function \((\phi,\psi)\colon\mathfrak{H}\times \mathfrak{H}\to\mathbb{C}\) as follows: if functions \(\phi,\psi\in\mathfrak{H}\) have representations

$$ \begin{alignedat}{3} \phi(t)&\equiv\sum_{k=1}^{n}a_k K(x_k,t),&\qquad n&\in\mathbb{N},&\quad \{x_k\}_{k=1}^n&\subset G, \\ \psi(t)&\equiv\sum_{p=1}^{m}b_p K(y_p,t),&\qquad m&\in\mathbb{N},&\quad \{y_p\}_{p=1}^m&\subset G, \end{alignedat}$$
(2.4)

then we set

$$ (\phi,\psi):=\sum_{k=1}^n\,\sum_{p=1}^m K(x_k,y_p)a_k\overline b_p= \sum_{k=1}^n a_k\overline{\psi(x_k)}= \sum_{p=1}^m \overline b_p\phi(y_p).$$
(2.5)

The function \((\phi,\psi)\colon\mathfrak{H}\times \mathfrak{H}\to\mathbb{C}\) is well defined, because it is independent of the choice of representations (2.4) for the functions \(\phi\) and \(\psi\). The function (2.5) is, obviously, a semi-inner product on \(\mathfrak{H}\), and hence

$$|(\phi,\psi)|^2\le (\phi,\phi)(\psi,\psi)\qquad \text{for all}\quad \phi,\psi\in\mathfrak{H}.$$

This is, in fact, inequality (2.2). It follows from this inequality that if \((\phi,\phi)=0\), then \((\phi,\psi)=0\) for any function \(\psi\in\mathfrak{H}\) and, in particular, for the function \(\psi(t)\equiv k_{\lambda}(t)\), where \(\lambda\) is an arbitrary element from \(G\). But \((\phi,k_{\lambda})=\phi(\lambda)\). Therefore, \(\phi\) is the zero element in \(\mathfrak{H}\). Thus, the function (2.5) is an inner product on \(\mathfrak{H}\). It follows that inequality (2.2) becomes an equality for some \(n,m\in\mathbb{N}\), \(\{x_k\}_{k=1}^n, \{y_p\}_{p=1}^m \subset G\), \(x\in G\), and \(\{a_k\}_{k=1}^n, \{b_p\}_{p=1}^m \subset \mathbb{C}\) if and only if, for all \(t\in G\), the following equality holds:

$$(\psi,\psi)\phi(t)-(\phi,\psi)\psi(t)=0,$$

where the functions \(\phi\) and \(\psi\) are from representation (2.4). Theorem 1 is proved.

Remark 2.

Another proof of inequality (2.2) for \(K\in\Phi(G\times G)\) can be obtained if for the linear space we take the set \(\mathfrak{L}\) of functions \(h\colon G\to\mathbb{C}\) that are zero everywhere on \(G\), except, possibly, at a finite number of points. Obviously, \(\mathfrak{L}\) is a linear space over the field \(\mathbb{C}\) with zero element \(h\equiv 0\) on \(G\). It is easy to verify that the function \((h,g)\colon\mathfrak{L}\times \mathfrak{L}\to\mathbb{C}\) is well defined and is a semi-inner product on \(\mathfrak{L}\):

$$ (h,g):=\sum_{x,y\in G}K(x,y)h(x)\overline{g(y)},\qquad h,g\in\mathfrak{L}.$$
(2.6)

Obviously, \(h\in\mathfrak{L}\) if and only if there exists a finite system of pairwise different points \(\{x_k\}_{k=1}^n \subset G\) such that \(h(x)=0\) for \(x\in G\setminus\{x_k\}_{k=1}^n\). If \(\{y_p\}_{p=1}^m\) is a similar system of pairwise different points from \(G\) for a function \(g\in\mathfrak{L}\), then

$$(h,g)=\sum_{k=1}^n\,\sum_{p=1}^m K(x_k,y_p) h(x_k)\overline{g(y_p)},\qquad (h,h)=\sum_{k,p=1}^n K(x_k,x_p)h(x_k)\overline{h(x_p)}\ge0.$$

Applying the Cauchy–Bunyakovskii inequality (2.1), we obtain inequality (2.2) but under the additional condition that \(x_k\ne x_p\) and \(y_k\ne y_p\) for \(k\ne p\). Now it is easy to verify the validity of inequality (2.2) without this additional condition. It is easy to show that the semi-inner product (2.6) is an inner product on \(\mathfrak{L}\) if and only if the kernel \(K\) is strictly positive definite.

Corollary 1.

Let \(K\in\Phi(G\times G)\) . Then \(K\) is not a strictly positive definite kernel if and only if there exists an \(n\in\mathbb{N}\) , and a collection \(\{x_k\}_{k=1}^{n}\subset G\) , in which \(x_k\ne x_p\) for \(k\ne p\) , such that the system of functions \(\{K(x_k,x)\}_{k=1}^{n}\) is linearly dependent on \(G\) .

Proof.

If \(K\in\Phi(G)\), then inequality (2.3) implies the equivalence

$$ \sum_{k,p=1}^{n} K(x_k,x_p) a_k\overline a_p=0\qquad\iff\qquad \sum_{k=1}^{n}a_k K(x_k,x)=0,\quad x\in G.$$
(2.7)

This yields the assertion of the corollary.

Corollary 2.

Let \(K\colon G\times G\to \mathbb{C}\). Then \(K\in\Phi(G\times G)\) if and only if \(K(x,x)\ge 0\) for all \(x\in G\) and, for all natural numbers \(n,m\in\mathbb{N}\), for arbitrary collections \(\{x_k\}_{k=1}^n,\{y_p\}_{p=1}^m \subset G\) and \(\{a_k\}_{k=1}^n,\{b_p\}_{p=1}^m \subset \mathbb{C}\), inequality (2.2) holds.

Proof.

Necessity has already been proved. Let us prove sufficiency. For \(n=m=1\), \(x_1=x\), \(y_1=y\), and \(a_1=b_1=1\), inequality (2.2) implies the inequality

$$|K(x,y)|^2\le K(x,x)K(y,y),\qquad x,y\in G.$$

Therefore, if \(K(x,x)=0\) for all \(x\in G\), then \(K\equiv 0\), and hence \(K\in\Phi(G\times G)\). If \(K(x,x)>0\) for some \(x\in G\), then from inequality (2.3), which follows from (2.2), we obtain inequality (1.1). Therefore, in this case, we also have \(K\in\Phi(G\times G)\).

Remark 3.

Corollary 2, obviously, remains valid if, instead of the validity of inequality (2.2), we require that inequality (2.3) holds. For functions \(f\in\Phi(G)\), these facts were noted in [8, Remarks, p. 25], where, however, the condition \(f(0)\ge 0\) was missing.

Theorem 1 can be written for \(f\in\Phi(G)\), where \(G\) is a group. Instead of a system \(\{x_k\}_{k=1}^{n}\subset G\), we may take the new system

$$\{\widetilde{x}_k=u+x_k+x\}_{k=1}^{n},\qquad\text{where}\quad u,x\in G.$$

Then

$$\widetilde{x}_k-\widetilde{x}_p=u+x_k+x+(-x-x_p-u)=u+x_k-x_p-u.$$

Let us write the result for \(u=0\).

Corollary 3.

Let \(G\) be a group, and let \(f\in\Phi(G)\). Then, for all natural numbers \(n,m\in\mathbb{N}\), for arbitrary collections \(\{x_k\}_{k=1}^n, \{y_p\}_{p=1}^m \subset G\), \(x\in G\), and \(\{a_k\}_{k=1}^n, \{b_p\}_{p=1}^m \subset \mathbb{C}\), the following inequality holds :

$$ \biggl|\sum_{k=1}^n\,\sum_{p=1}^m f(x_k+x-y_p) a_k\overline b_p\biggr|^2\le \sum_{k,p=1}^n f(x_k-x_p)a_k\overline a_p \sum_{k,p=1}^m f(y_k-y_p)b_k\overline b_p.$$
(2.8)

Here inequality (2.8) becomes an equality for some natural numbers \(n,m\in\mathbb{N}\), collections \(\{x_k\}_{k=1}^n,\{y_p\}_{p=1}^m \subset G\), \(x\in G\), and \(\{a_k\}_{k=1}^n, \{b_p\}_{p=1}^m \subset \mathbb{C}\) if and only if, for all \(t\in G\), the following equality holds :

$$\sum_{k,p=1}^m f(y_k-y_p)b_k\overline b_p \sum_{k=1}^{n}a_k f(x_k+x-t)= \sum_{k=1}^n\,\sum_{p=1}^m f(x_k+x-y_p)a_k\overline b_p \sum_{p=1}^{m}b_p f(y_p-t).$$

If, in (2.8), we take \(m=n\), \(y_k=x_k\), and \(b_k=a_k\) for \(k=1,\dots,n\), then we obtain the following corollary.

Corollary 4.

Let \(G\) be a group, and let \(f\in\Phi(G)\). Then, for all \(n\in\mathbb{N}\), \(\{x_k\}_{k=1}^n \subset G\), \(x\in G\) and \(\{a_k\}_{k=1}^n\subset \mathbb{C}\), the following inequality holds :

$$ \biggl|\sum_{k,p=1}^n f(x_k+x-x_p)a_k\overline a_p\biggr| \le\sum_{k,p=1}^n f(x_k-x_p)a_k\overline a_p.$$
(2.9)

Inequality (2.9) becomes an equality for some \(n\in\mathbb{N}\), \(\{x_k\}_{k=1}^n \subset G\), \(x\in G\), and \(\{a_k\}_{k=1}^n \subset \mathbb{C}\) if and only if, for all \(t\in G\), the following equality holds:

$$\sum_{k,p=1}^n f(x_k-x_p)a_k\overline a_p \sum_{k=1}^{n}a_k f(x_k+x-t)= \sum_{k,p=1}^n f(x_k+x-x_p)a_k\overline a_p \sum_{p=1}^{n}a_p f(x_p-t).$$

Remark 4.

Inequality (2.9) is equivalent to the inequality \(|\psi(x)|\le \psi(0)\), where

$$\psi(x):=\sum_{k,p=1}^{n}f(x_k+x-x_p)a_k\overline a_p,\qquad x\in G.$$

Since \(f\in\Phi(G)\), it is easy to verify that \(\psi\in\Phi(G)\). Therefore, the inequality \(|\psi(x)|\le \psi(0)\) becomes an equality for some \(x\in G\) if and only if, for all \(t\in G\), the following equality holds:

$$\psi(0)\psi(x+t)=\psi(x)\psi(t).$$

The useful inequality for \(f\in\Phi(G)\) is obtained when \(G\) is an Abelian group, \(n=2\), \(a_1=1\), \(a_2=-1\), \(x_1=h\), and \(x_2=0\):

$$|2f(x)-f(x+h)-f(x-h)|\le 2\operatorname{Re}(f(0)-f(h)),\qquad x,h\in G$$

(see, e.g., [11, 6.2.1 (c)] and [12, (5)]).

We note that if a function \(f\colon G\to \mathbb{C}\) is a character on \(G\), i.e.,

$$|f(x)|=1,\qquad f(x+y)=f(x)f(y),\quad x,y\in G,$$

then \(f\in\Phi(G)\) and, for it, inequality (2.8) becomes an equality.

Corollary 5.

Let \(f\in\Phi(\mathbb{R}^d)\) , and let \(f(x)=0\) for all \(x\in\mathbb{Z}^d\setminus\{0\}\) . Let \(X= \{x_k\}_{k=1}^{\infty}\) and \(Y=\{y_p\}_{p=1}^{\infty}\) be two systems each of which consists of a countable number of pairwise different points from \(\mathbb{Z}^d\) . Then, for all \(n,m\in\mathbb{N}\) , \(x\in\mathbb{R}^d\) , and all collections \(\{a_k\}_{k=1}^{n}\) and \(\{b_p\}_{p=1}^{m}\) of numbers from \(\mathbb{C}\) , the following inequality holds :

$$ \biggl|\sum_{k=1}^n\,\sum_{p=1}^m f(x+x_k-y_p) a_k\overline b_p\biggr|\le f(0)\cdot \sqrt{\,\sum_{k=1}^{n}|a_k|^2}\cdot\sqrt{\,\sum_{p=1}^{m}|b_p|^2}\,.$$
(2.10)

In particular, if \(X=\{x_k\}_{k=1}^{\infty}\) and \(Y=\{y_p\}_{p=1}^{\infty}\) are two systems each of which consists of a countable number of pairwise different points from \(\mathbb{Z}\) , then, for all \(n,m\in\mathbb{N}\) , \(x\in\mathbb{R}\) , and all collections \(\{\alpha_k\}_{k=1}^{n}\) and \(\{\beta_p\}_{p=1}^{m}\) of numbers from \(\mathbb{C}\) , the following inequality holds :

$$ \biggl|\sum_{k=1}^n\,\sum_{p=1}^m\frac{\sin(\pi x)\alpha_k\beta_p} {x+x_k-y_p}\biggr|\le\pi\cdot \sqrt{\,\sum_{k=1}^{n}|\alpha_k|^2} \cdot \sqrt{\,\sum_{p=1}^{m}|\beta_p|^2}\,.$$
(2.11)

Proof.

Inequality (2.10), obviously, follows from inequality (2.8) for \(G=\mathbb{R}^d\). Inequality (2.11) is obtained from inequality (2.10) for \(d=1\) if the positive definite function \(f(x)=\sin(\pi x)/x\) on \(\mathbb{R}\), \(a_k:=(-1)^{x_k}\alpha_k\), and \(b_p:=(-1)^{y_p}\overline\beta_p\) are taken.

Remark 5.

If, in inequality (2.11), we take \(X=\mathbb{Z}_+\), \(Y=-\mathbb{Z}_+\), \(x>0\), \(\alpha_k,\beta_p\ge 0\), \(p,k\in\mathbb{Z}_+\), and \(\{\alpha_k\}_{k=0}^{\infty}, \{\beta_p\}_{p=0}^{\infty}\in \ell_2\), then, by passing to the limit, we obtain the following Ingham inequality (see, e.g., [13] or [14, Complement 42]):

$$ \sum_{k,p=0}^{\infty}\frac{\alpha_k\beta_p}{x+k+p}\le M(x)\cdot \sqrt{\,\sum_{k=0}^{\infty}\alpha_k^2} \cdot \sqrt{\,\sum_{p=0}^{\infty}\beta_p^2}\,,$$
(2.12)

where \(M(x)=\pi/\sin(\pi x)\) for \(0<x\le 1/2\) and \(M(x)=\pi\) for \(x\ge 1/2\). The constant \(M(x)\) in inequality (2.12) is exact. If both sequences \(\{\alpha_k\}_{k=0}^{\infty}\) and \(\{\beta_p\}_{p=0}^{\infty}\) are nonzero, then, for \(x\ge 1/2\), inequality (2.12) is strict, and for \(0<x< 1/2\), the equality is attained for

$$\alpha_k=\beta_k=\frac{\Gamma(k+x)}{\Gamma(k+1)}\mspace{2mu},\qquad k\in\mathbb{Z}_+.$$

For more detail, see [13]; see also Sec. 8.8 (for \(p=q=2\)) and Supplement 42 from the Russian translation of the monograph [14]. We note that, for \(x=1/2\), the well-known Hilbert inequality is obtained.

For \(n=2\), \(m=1\), \(y_1=x\), \(a_1=\alpha\), \(a_2=-\beta\), and \(b_1=1\), from inequality (2.2), we immediately obtain inequality (2.13), which, for \(\alpha=\beta=1\), is Krein’s inequality (1.3).

Corollary 6.

Let \(K\in\Phi(G\times G)\). Then, for all \(x_1,x_2,x\in G\) and \(\alpha,\beta\in\mathbb{C}\), the following inequality holds :

$$|\alpha K(x_1,x)-\beta K(x_2,x)|^2 \le K(x,x)\operatorname{Re}(|\alpha|^2 K(x_1,x_1)+ |\beta|^2 K(x_2,x_2)-2\alpha\overline{\beta}K(x_1,x_2)). $$
(2.13)

Inequality (2.13) becomes an equality for some \(x_1,x_2,x\in G\) and \(\alpha,\beta\in\mathbb{C}\) if and only if, for all \(t\in G\), the following equality holds :

$$K(x,x)(\alpha K(x_1,t)-\beta K(x_2,t))= (\alpha K(x_1,x)-\beta K(x_2,x))K(x,t).$$

Corollary 7.

Let \(K\in\Phi(G\times G)\). Then, for all \(x,y,u\in G\), the following inequality holds :

$$\begin{aligned} \, \nonumber &|K(u,u)K(x,y)-K(x,u)K(u,y)|^2 \\ &\qquad \le(K(u,u)K(x,x)-|K(x,u)|^2)(K(u,u)K(y,y)-|K(u,y)|^2). \end{aligned}$$
(2.14)

Inequality (2.14) becomes an equality for some \(x,y,u\in G\) if and only if, for all \(t\in G\), the following equality holds :

$$\begin{aligned} \, \nonumber &(K(y,y)K(u,u)-K(y,u)K(u,y))\cdot (K(x,u)K(u,t)-K(u,u)K(x,t)) \\ &\qquad =(K(x,y)K(u,u)-K(x,u)K(u,y))\cdot (K(y,u)K(u,t)-K(u,u)K(y,t)). \end{aligned}$$
(2.15)

Remark 6.

If \(K(x,y)=f(x-y)\), where \(f\in\Phi(G)\), and \(G\) is a group, then inequality (2.14) is Weil’s inequality (see, e.g., [15, Sec. 14]).

Proof of Corollary 7 .

Let us write (2.2) for \(n=m=2\), \(x_1=y_1=u\), \(x_2=x\), \(y_2=y\), \(a_1=K(x,u)\), \(a_2=-K(u,u)\), \(b_1=K(y,u)\), and \(b_2=-K(u,u)\). We obtain the inequality

$$\begin{aligned} \, \nonumber &K^2(u,u)|K(u,u)K(x,y)-K(x,u)K(u,y)|^2 \\ &\qquad \le K^2(u,u)(K(u,u)K(x,x)-|K(x,u)|^2) (K(u,u)K(y,y)-|K(u,y)|^2), \end{aligned}$$
(2.16)

which is equivalent to inequality (2.14). For \(K(u,u)>0\), this is obvious. If \(K(u,u)=0\), then \(K(u,y)=0\) for all \(y\in G\) (because \(|K(u,y)|^2\le K(u,u)K(y,y)\), \(u,y\in G\)) and both inequalities (2.14) and (2.16) are trivial. In the same way, (2.14) becomes an equality if and only if (2.16) becomes an equality, which is equivalent to the validity, for all \(t\in G\), of the equalities

$$\begin{aligned} \, &(K(y,y)K(u,u)-K(y,u)K(u,y))\cdot (K(x,u)K(u,t)-K(u,u)K(x,t))\cdot K(u,u) \\&\qquad =(K(x,y)K(u,u)-K(x,u)K(u,y))\cdot (K(y,u)K(u,t)-K(u,u)K(y,t))\cdot K(u,u). \end{aligned}$$

The last identity is equivalent to identity (2.15).

Corollary 8.

Let \(K\in\Phi(G\times G)\) , and let \(\mathbb{D}:=\{z\in\mathbb{C}:|z|\le 1\}\) . Then, for all \(n,m\in\mathbb{N}\) , for all collections \(\{x_k\}_{k=1}^n,\{y_p\}_{p=1}^m \subset G\) , all points \(x,y,u\in G\) , and for all \(\{a_k\}_{k=1}^n,\{b_p\}_{p=1}^m \subset {\mathbb{D}}\) and \(\alpha,\beta,\xi_1,\xi_2\in\mathbb{D}\) , the following inequalities hold :

$$\begin{aligned} \, \biggl|\sum_{k=1}^n\,\sum_{p=1}^m K(x_k,y_p) a_k\overline b_p\biggr|^2 &\le \operatorname{Re}\biggl(\,\sum_{k=1}^{n}K(x_k,x_k)+ 2\sum_{1\le k<p\le n} K(x_k,x_p)a_k\overline a_p\biggr) \\ &\qquad\times \operatorname{Re}\biggl(\,\sum_{p=1}^{m}K(y_p,y_p)+ 2\sum_{1\le k<p\le m} K(y_k,y_p)b_k\overline b_p\biggr), \end{aligned}$$
(2.17)
$$\biggl|\sum_{k=1}^n a_k K(x_k,x)\biggr|^2\le K(x,x)\operatorname{Re}\biggl(\,\sum_{k=1}^{n}K(x_k,x_k)+ 2\sum_{1\le k<p\le n}K(x_k,x_p)a_k\overline a_p\biggr),$$
(2.18)
$$|\alpha K(x_1,x)-\beta K(x_2,x)|^2\le K(x,x)\operatorname{Re}(K(x_1,x_1)+K(x_2,x_2)- 2\alpha \overline{\beta} K(x_1,x_2)),$$
(2.19)
$$\begin{aligned} \, &|K(u,u)+\xi_1\xi_2K(x,y)-\xi_1 K(x,u)-\xi_2 K(u,y)|^2 \\ &\qquad \le 4\operatorname{Re} \biggl(\frac{K(u,u)+K(x,x)}{2}-\xi_1 K(x,u)\biggr) \operatorname{Re}\biggl(\frac{K(u,u)+K(y,y)}{2}-\xi_2K(u,y)\biggr). \end{aligned}$$
(2.20)

Remark 7.

For positive definite functions on a group \(G\), inequality (2.19) is Gorin’s inequality [4] (for \(G=\mathbb{R}\), see also the paper [7]), and it is one of the generalizations of Krein’s inequality.

Proof.

Inequality (2.17) follows from inequality (2.2) and the following inequalities, which hold for all \(\{a_k\}_{k=1}^n,\{b_p\}_{p=1}^m \subset {\mathbb{D}}\):

$$\begin{aligned} \, 0&\le \sum_{k,p=1}^n K(x_k,x_p)a_k\overline a_p= \sum_{k=1}^{n}K(x_k,x_k)|a_k|^2+2\operatorname{Re} \sum_{1\le k<p\le n} K(x_k,x_p)a_k\overline a_p \\& \le\sum_{k=1}^{n}K(x_k,x_k)+2\operatorname{Re} \sum_{1\le k<p\le n} K(x_k,x_p)a_k\overline a_p, \\ 0&\le\sum_{k,p=1}^m K(y_k,y_p)b_k\overline b_p\le \sum_{k=1}^{m}K(y_k,y_k)+2\operatorname{Re} \sum_{1\le k<p\le m} K(y_k,y_p)b_k\overline b_p. \end{aligned}$$

Inequality (2.18) follows from inequality (2.17) for \(m=1\), \(y_1=x\), and \(b_1=1\).

For \(\alpha,\beta\in\mathbb{D}\), inequality (2.19) follows from inequality (2.18) for \(n=2\), \(a_1=\alpha\), and \(a_2=-\beta\).

For \(\xi_1,\xi_2\in\mathbb{D}\), inequality (2.20) follows from (2.17) for \(n=m=2\), \(x_1=y_1=u\), \(x_2=x\), \(y_2=y\), \(a_1=1\), \(a_2=-\xi_1\), \(b_1=1\), and \(b_2=-\overline\xi_2\). It should be noted that \(K(x,x)\ge 0\) and \(K(u,x)=\overline{K(x,u)}\) for all \(u,x\in G\).

3. Inequalities of the type of the Krein–Gorin inequality

Theorem 2.

Let \(K\in\Phi(G\times G)\) . Then, for any \(n\in\mathbb{N}\) , for arbitrary \(\{x_k\}_{k=1}^{n+1} \subset G\) and \(u\in G\) , and for all \(\xi_k,\eta_k\in\mathbb{C}\) , \(|\xi_k|\le 1\) , \(|\eta_k|\le 1\) , \(k=1,\dots,n\) , the following inequalities hold :

$$\begin{aligned} \, &\sqrt{\biggl|\frac{K(x_{1},x_{1})+K(x_{n+1},x_{n+1})}{2}- \xi_1\cdots\xi_n K(x_{1},x_{n+1})\biggr|} \end{aligned}$$
$$\begin{aligned} \, &\qquad \le\sum_{k=1}^{n} \sqrt{\biggl|\frac{K(x_{k},x_{k})+K(x_{k+1},x_{k+1})}{2}- \xi_k K(x_{k},x_{k+1})\biggr|}\,, \end{aligned}$$
(3.1)
$$\begin{aligned} \, &\sqrt{\operatorname{Re} \biggl(\frac{K(x_{1},x_{1})+K(x_{n+1},x_{n+1})}{2}- \xi_1\cdots\xi_n K(x_{1},x_{n+1})\biggr)} \end{aligned}$$
$$\begin{aligned} \, &\qquad \le\sum_{k=1}^{n}\sqrt{\operatorname{Re} \biggl(\frac{K(x_{k},x_{k})+K(x_{k+1},x_{k+1})}{2}- \xi_k K(x_{k},x_{k+1})\biggr)}\,, \end{aligned}$$
(3.2)
$$\begin{aligned} \, &|\xi_1\cdots\xi_n K(x_1,u)-\eta_1\cdots\eta_n K(x_{n+1},u)| \end{aligned}$$
$$\begin{aligned} \, &\qquad \le\sum_{k=1}^{n}\sqrt{2K(u,u)\operatorname{Re} \biggl(\frac{K(x_{k},x_{k})+K(x_{k+1},x_{k+1})}{2}- \xi_k \overline{\eta_k} K(x_{k},x_{k+1})\biggr)}\,. \end{aligned}$$
(3.3)
$$\begin{aligned} \, &|\xi_1\cdots\xi_n K(x_1,u)-\eta_1\cdots\eta_n K(x_{n+1},u)|^2 \end{aligned}$$
$$\begin{aligned} \, &\qquad \le 2n\,K(u,u)\sum_{k=1}^{n}\operatorname{Re} \biggl(\frac{K(x_{k},x_{k})+K(x_{k+1},x_{k+1})}{2}- \xi_k \overline{\eta_k} K(x_{k},x_{k+1})\biggr). \end{aligned}$$
(3.4)

Proof.

It, obviously, suffices to prove inequalities (3.1) and (3.2) for \(n=2\). Let \(|\xi_1|\le 1\), \(|\xi_2|\le 1\), \(x,y,u\in G\), and let

$$\begin{gathered} \, I:=\frac{K(x,x)+K(y,y)}{2}-\xi_1\xi_2 K(x,y),\qquad I_1:=\frac{K(x,x)+K(u,u)}{2}-\xi_1 K(x,u), \\ I_2:=\frac{K(u,u)+K(y,y)}{2}-\xi_2 K(u,y), \\ I_3:= K(u,u)+\xi_1\xi_2 K(x,y)-\xi_1 K(x,u)-\xi_2 K(u,y). \end{gathered}$$

Obviously, \(I=I_1+I_2-I_3\). Since

$$\begin{gathered} \, \operatorname{Re}(\xi K(u,v))\le |K(u,v)|\le \sqrt{K(u,u)}\,\sqrt{K(v,v)}\le \frac{K(u,u)+K(v,v)}{2}\mspace{2mu}, \\ |\xi|\le 1,\quad u,v\in G, \end{gathered}$$

it follows that \(\operatorname{Re}I\ge 0\), \(\operatorname{Re}I_1\ge 0\), and \(\operatorname{Re}I_2\ge 0\). Applying the triangle inequality and inequality (2.20), we obtain

$$\begin{aligned} \, |I|&\le |I_1|+|I_2|+|I_3|\le |I_1|+|I_2|+2\sqrt{\operatorname{Re} I_1}\,\sqrt{\operatorname{Re}I_2} \\& \le |I_1|+|I_2|+2\sqrt{|I_1|}\,\sqrt{|I_2|}= (\sqrt{|I_1|}+\sqrt{|I_2|}\,)^2. \end{aligned}$$

For \(n=2\), \(x_1=x\), \(x_2=u\), and \(x_3=y\), inequality (3.1) is proved. Similarly, we prove inequality (3.2):

$$\begin{aligned} \, 0&\le \operatorname{Re}I=\operatorname{Re}I_1+ \operatorname{Re}I_2-\operatorname{Re}I_3\le \operatorname{Re}I_1+\operatorname{Re}I_2+|I_3| \\& \le\operatorname{Re}I_1+\operatorname{Re}I_2+ 2\sqrt{\operatorname{Re}I_1}\,\sqrt{\operatorname{Re}I_2}= (\sqrt{\operatorname{Re}I_1}+\sqrt{\operatorname{Re}I_2}\,)^2. \end{aligned}$$

For \(n=2\), \(x_1=x\), \(x_2=u\), and \(x_3=y\), inequality (3.2) is also proved.

Applying inequalities (2.19) and (3.2) one after the other, we obtain

$$\begin{aligned} \, &|\xi_1\cdots\xi_n K(x_1,u)-\eta_1\cdots\eta_n K(x_{n+1},u)|^2 \\&\qquad \le 2K(u,u)\operatorname{Re} \biggl(\frac{K(x_{1},x_{1})+K(x_{n+1},x_{n+1})}{2}- \xi_1\overline\eta_1\cdots \xi_n\overline\eta_n K(x_{1},x_{n+1})\biggr) \\&\qquad \le 2K(u,u)\biggl(\,\sum_{k=1}^{n} \sqrt{\operatorname{Re} \biggl(\frac{K(x_{k},x_{k})+K(x_{k+1},x_{k+1})}{2}- \xi_k \overline\eta_k K(x_{k},x_{k+1})}\biggr)^2. \end{aligned}$$

Inequality (3.3) is proved. Applying the Cauchy–Bunyakovskii inequality to the right-hand side of inequality (3.3), we obtain inequality (3.4). The theorem is proved.

Remark 8.

Let \(G\) be a group (not necessarily Abelian), and let \(f\in\Phi(G)\). In Theorem 2, we take \(K(x,y)=f(x-y)\) and \(u=0\). Then, for any \(n\in\mathbb{N}\), for arbitrary collection \(\{x_k\}_{k=1}^{n+1} \subset G\) of points, and for all numbers \(\xi_k,\eta_k\in\mathbb{C}\), \(|\xi_k|\le 1\), \(|\eta_k|\le 1\), \(k=1,\dots,n\), the following inequalities hold:

$$\sqrt{|f(0)-\xi_1\cdots\xi_n f(x_{1}-x_{n+1})|} \le \sum_{k=1}^{n}\sqrt{|f(0)-\xi_k f(x_{k}-x_{k+1})|}\,,$$
(3.5)
$$\sqrt{\operatorname{Re}(f(0)-\xi_1\cdots\xi_n f(x_{1}-x_{n+1}))} \le\sum_{k=1}^{n} \sqrt{\operatorname{Re}(f(0)-\xi_k f(x_{k}-x_{k+1}))}\,,$$
(3.6)
$$|\xi_1\cdots\xi_n f(x_1)-\eta_1\cdots\eta_n f(x_{n+1})| \le \sum_{k=1}^{n}\sqrt{2 f(0)\operatorname{Re} (f(0)-\xi_k \overline{\eta_k}f(x_k-x_{k+1}))}\,,$$
(3.7)
$$|\xi_1\cdots\xi_n f(x_1)-\eta_1\cdots\eta_n f(x_{n+1})|^2 \le 2n f(0)\sum_{k=1}^{n}\operatorname{Re}(f(0)- \xi_k \overline{\eta_k}f(x_k-x_{k+1})).$$
(3.8)

If \(\{u_k\}_{k=1}^n\) is a collection of arbitrary points from \(G\), then, in inequalities (3.5) and (3.6), we may take \(x_{k+1}:=-u_k+x_k\), \(k=1,\dots,n\). In that case,

$$x_{k}-x_{k+1}=u_k,\quad k=1,\dots,n,\qquad x_1-x_{n+1}=u_1+\cdots+u_n.$$

If \(\{v_k\}_{k=1}^n\) is also a system of arbitrary points from \(G\), then, in inequalities (3.7) and (3.8), we may take \(x_{k+1}:=v_k-u_k+x_k\), \(k=1,\dots,n\). Then

$$x_{k}-x_{k+1}=u_k-v_k,\quad k=1,\dots,n,\qquad x_{n+1}=-((u_1-v_1)+\cdots+(u_n-v_n))+x_1.$$

If, additionally, \(G\) is an Abelian group and \(x_1:=u_1+\cdots+u_n\), then \(x_{n+1}=v_1+\cdots+v_n\), and, for \(f\in\Phi(G)\) and for all numbers \(\xi_k,\eta_k\in\mathbb{C}\), \(|\xi_k|\le 1\), \(|\eta_k|\le 1\), \(k=1,\dots,n\), inequalities (3.7) and (3.8) take the following form:

$$\begin{gathered} \, \nonumber \biggl|\xi_1\cdots\xi_n f\biggl(\,\sum_{k=1}^{n}u_k\biggr)- \eta_1\cdots\eta_n f\biggl(\,\sum_{k=1}^{n}v_k\biggr)\biggr| \le \sum_{k=1}^{n}\sqrt{2f(0)\operatorname{Re} (f(0)-\xi_k \overline{\eta_k}f(u_k-v_k))}\,, \\ \biggl|\xi_1\cdots\xi_n f\biggl(\,\sum_{k=1}^{n}u_k\biggr)- \eta_1\cdots\eta_n f\biggl(\,\sum_{k=1}^{n}v_k\biggr)\biggr|^2 \le 2n f(0)\sum_{k=1}^{n}\operatorname{Re} (f(0)-\xi_k \overline{\eta_k} f(u_k-v_k)). \end{gathered}$$
(3.9)

For \(\xi_k=\xi\), \(\eta_k=\eta\), \(k=1,\dots,n\), \(|\xi|\le 1\), and \(|\eta|\le 1\), inequality (3.9) was proved by Gorin [4] in 2015. The case \(\xi=\eta=1\) was obtained earlier in [5], [6] (see also the paper [7] in which, for \(G=\mathbb{R}\), this case and the case \(\xi=1\), \(\eta=-1\) were studied). In these papers, in contrast to our method, Bochner’s theorem was used.

4. Inequalities of Ingham type

Let \(G\) be a group. By the Minkowski difference of two nonempty sets \(X,Y\subset G\) we mean

$$X-Y:=\{x-y:x\in X,y\in Y\}.$$

Let \(\Lambda(X):=X-X\). Obviously, the set \(\Lambda(X)\) is symmetric about zero and \(0\in\Lambda(X)\).

Theorem 3.

Let \(f\in \Phi(G)\) , let \(n,m\in\mathbb{N}\) , and let \(X=\{x_k\}_{k=1}^{n}\) and \(Y=\{y_p\}_{p=1}^{m}\) be two systems each of which consists of a finite number of pairwise different points from \(G\) . Then, for any \(x\in G\) and for all collections \(\{a_k\}_{k=1}^{n}\) and \(\{b_p\}_{p=1}^{m}\) of numbers from \(\mathbb{C}\) , the following inequalities hold :

$$\biggl|\,\sum_{k=1}^n\,\sum_{p=1}^m f(x_k+x-y_p) a_k\overline b_p\biggr|^2 \le \sum_{u\in \Lambda(X)} |f(u)| \cdot \sum_{v\in \Lambda(Y)}|f(v)| \cdot \sum_{k=1}^{n}|a_k|^2 \cdot \sum_{p=1}^{m}|b_p|^2,$$
(4.1)
$$\sum_{p=1}^m\biggl|\,\sum_{k=1}^n f(x_k+x-y_p)a_k\biggr|^2 \le \sum_{u\in \Lambda(X)}|f(u)| \cdot \sum_{v\in \Lambda(Y)}|f(v)|\cdot \sum_{k=1}^{n}|a_k|^2.$$
(4.2)

Remark 9.

Inequality (4.1) is a generalization of inequality (2.10) which, in turn, implies Ingham’s inequality (2.12). Therefore,inequality (4.1) will be called an inequality of Ingham type.

First, let us prove the following lemma.

Lemma 1.

Let \(f\colon G\to\mathbb{C}\) , let \(n\in\mathbb{N}\) , and let \(X=\{x_k\}_{k=1}^{n}\) be a finite system of pairwise different points from \(G\) . Then, for any collection \(\{a_k\}_{k=1}^{n}\subset\mathbb{C}\) , the following inequality holds :

$$ \biggl|\,\sum_{k,p=1}^{n}f(x_k-x_p)a_k\overline a_p\biggr|\le \sum_{u\in \Lambda(X)}|f(u)| \cdot \sum_{k=1}^{n}|a_k|^2.$$
(4.3)

Proof.

Let \(\Lambda(X)=\{u_1,\dots,u_q\}\), where \(u_s\ne u_p\) for \(s\ne p\), and let \(q:=|\Lambda(X)|\) be the number of elements in \(\Lambda(X)\). For any fixed \(k\in\{1,\dots,n\}\) and \(s\in\{1,\dots,q\}\), if the equation \(x_k-x_p=u_s\) has a solution for \(p\in\{1,\dots,n\}\), then it is unique. Similarly, for any fixed \(p\in\{1,\dots,n\}\) and \(s\in\{1,\dots,q\}\), if the equation \(x_k-x_p=u_s\) has a solution for \(k\in\{1,\dots,n\}\), then it is unique. Therefore, each set

$$\Lambda_s:=\{(k,p):x_k-x_p=u_s,\,k,p\in\{1,\dots,n\}\},\qquad s\in\{1,\dots,q\},$$

can be represented as \(\Lambda_s=\{(k(s,j),p(s,j))\}_{j=1}^{m(s)}\), where \(m(s)=|\Lambda_s|\in\{1,\dots,n\}\), \(k(s,j)\ne k(s,l)\), and \(p(s,j)\ne p(s,l)\) for \(j\ne l\). Then

$$\begin{aligned} \, \biggl|\sum_{k,p=1}^{n}f(x_k-x_p)a_k\overline a_p\biggr|&\le \sum_{s=1}^{q}|f(u_s)|\sum_{(k,p)\in\Lambda_s}|a_k|\,|a_p|= \sum_{s=1}^{q}|f(u_s)|\sum_{j=1}^{m(s)}|a_{k(s,j)}|\,|a_{p(s,j)}| \\& \le\sum_{s=1}^{q} |f(u_s)|\cdot \sqrt{\sum_{j=1}^{m(s)}|a_{k(s,j)}|^2}\cdot \sqrt{\sum_{j=1}^{m(s)}|a_{p(s,j)}|^2} \le\sum_{s=1}^{q} |f(u_s)| \cdot\sum_{k=1}^{n}|a_k|^2. \square \end{aligned}$$

Proof of Theorem 3 .

Inequality (4.1) follows from inequalities (2.8) and (4.3). Inequality (4.2) follows from inequality (4.1).

Theorem 4.

Let \(f\in\Phi(G)\) , let \(n\in\mathbb{N}\cup \{\infty\}\) , and let \(X=\{x_k\}_{k=1}^{n}\) be a finite or countable system of pairwise different points from \(G\) , i.e., \(x_k\in G\) and \(x_k\ne x_p\) for \(k\ne p\) . Then, for any \(x\in G\) , the following inequality holds :

$$ \sum_{k=1}^{n}|f(x_k+x)|^2\le f(0)\sum_{u\in\Lambda(X)}|f(u)|.$$
(4.4)

Proof of Theorem 4 .

Let \(f\in\Phi(G)\), and let \(n\in\mathbb{N}\). Applying inequality (2.8) for \(m=1\), \(y_1=0\), and \(b_1=1\) and Lemma 1, we see that, for any \(x\in G\) and for any collection \(\{a_k\}_{k=1}^{n}\subset\mathbb{C}\), the following inequality holds:

$$\biggl|\,\sum_{k=1}^n a_k f(x_k+x)\biggr|^2\le f(0)\sum_{k,p=1}^n f(x_k-x_p)a_k\overline a_p\le f(0) \sum_{u\in \Lambda(X)} |f(u)| \cdot \sum_{k=1}^{n}|a_k|^2.$$

Since the numbers \(a_k\) are arbitrary, we obtain inequality (4.4).

The case \(n=\infty\) is obtained from the proved case by passing to the limit as \(n\to\infty\) in inequality (4.4). It should be noted that if \(X=\{x_k\}_{k=1}^{\infty}\) is a countable system of pairwise different points and \(X_n:=\{x_k\}_{k=1}^{n}\), then \(\Lambda(X_n)\subset\Lambda(X)\) for any \(n\in\mathbb{N}\).