1. Introduction
We first present the classical Shannon’s entropy; see [1, 2]. Let X be a discrete random variable taking values in A, and let p(xi)=Pr(X=xi) be the probability mass function of X; then, the Shannon entropy of X is defined by
(1)H(X)∶=-∑xi∈Ap(xi)log(p(xi)).
We continue with some basic theoretical information about graphs; see [3, 4]. We consider G=(V,E), |V|<∞ to be a finite undirected graph, where E⊆(V2). G is connected if for any arbitrary vertices vi, vj exists an undirected path from vi to vj. Furthermore, we will consider 𝒢UC to be the set of finite undirected and connected graphs. This is usually used in strategies for improving transport efficiency in Network Information Content, including designing efficient routing strategies and making appropriate adjustments to the underlying network structure [5].

Now, taking into account the work presented in [6, 7], we define the concept of information functional. Let G=(V,E)∈𝒢UC, and let S initially be an abstract set; then, f:S→R+ is called the information functional of G (we assume that f is monotonous). The abstract set S can be, for example, the vertex sets, a set of paths, certain subgraphs, and so forth. The set S is used to define the functional f that captures the structural information of graph G and has to be defined concretely.

Using different information functionals to measure the entropy of a graph, we obtain different probability distributions; so, the resulting graph entropies are also different. Further, we define, as in [6, 7], the entropy of a graph G=(V,E)∈𝒢UC with arbitrary vertex labels vi, using an arbitrary information functional f, to be
(2)If(G)∶=-∑i=1|V|f(vi)∑j=1|V|f(vj)log(f(vi)∑j=1|V|f(vj)),
and considering a vertex vi∈V, pf(vi)=f(vi)/∑j=1|V|f(vj) interpreted as vertex probabilities (∑i=1|V|pf(vi)=1), we conclude that
(3)If(G)=-∑i=1|V|pf(vi)log(pf(vi)).

2. Refinement of a Classical Logarithmic Inequality
It is well known from the literature that
(4)logb(x)≤1ln(b)(x-1), ∀x>0.
Trying to refine this inequality, we consider the classical discrete Bernoulli inequality, as follows.

Theorem 1 (Bernoulli inequality).
Let ai≥0, xi>-1, i=1,…,n and ∑i=1nai≤1. Then,
(5)∏i=1n(1+xi)ai≤1+∑i=1naixi,
if ai≥1 or ai≤0, and if xi>0, or -1<xi<0, i=1,…,n, then
(6)∏i=1n(1+xi)ai≥1+∑i=1naixi.
Now, using a factorization of a real number, one concludes as follows.

Theorem 2.
Let x>0 be a real number; then,
(7)logb(x)≤1ln(b)∑i=1nxi≤1ln(b)(x-1),
where n≥2 and x=∏i=1n(xi+1), with xi>0 if x>1, xi=0 if x=1, and 0>xi>-1 if 1>x>0, ∀1≤i≤n.

Proof.
The inequality becomes an identity for x=1; that is, xi=0, ∀1≤i≤n. We will treat the other two cases, when x>1 and when 1>x>0 in the same manner, as follows:
(8)logb(x)=logb∏i=1n(xi+1)=∑i=1nlogb(xi+1),
for which applying the classical logarithmic inequality for xi+1>0, ∀1≤i≤n yields
(9)logb(x)≤1ln(b)∑i=1nxi,
and making use of the second part of the discrete Bernoulli inequality, previously presented, we obtain
(10)1ln(b)∑i=1nxi≤1ln(b)[∏i=1n(xi+1)-1]=1ln(b)(x-1),
and we are done.

3. Refinement of Dehmer and Mowshowitz Inequalities between Information Measures
In the work of Dehmer and Mowshowitz [6], by Theorems 4.7 and 4.8 are presented two information inequalities derived assuming only the characteristic properties of the functions involved. Here, we refine those results by using the previous theorem, as follows.

Theorem 3.
Let f and f* be information functionals. Then (all logarithms are in base 2),
(11)If(G)≥-∑k=1|V|pf(vk)log(pf*(vk))-1ln(2)∑i=1npik∑k=1|V|pf(vk)≥-∑k=1|V|pf(vk)log(pf*(vk))-1ln(2)∑k=1|V|(pf(vk))2-pf(vk)pf*(vk)pf*(vk),If*(G)≤1ln(2)∑i=1npik∑k=1|V|pf*(vk) -∑k=1|V|pf*(vk)log(pf(vk))≤1ln(2)∑k=1|V|[(pf(vk))pf(vk)-pf*(vk) -ln(2)pf*(vk)log(pf(vk))],
where n≥2 and pf(vk)/pf*(vk)=∏i=1n(pik+1), ∀1≤k≤|V|, with ∀1≤k≤|V|, pik>0 if pf(vk)>pf*(vk), pik=0 if pf(vk)=pf*(vk), and 0>pik>-1 if pf(vk)<pf*(vk), ∀1≤i≤n.

Proof.
Applying Theorem 2 for x=pf(vk)/pf*(vk)>0 yields that
(12)log(pf(vk))-log(pf*(vk)) ≤1ln(2)∑i=1npik ≤1ln(2)pf(vk)-pf*(vk)pf*(vk), ∀1≤k≤|V|.
By multiplying this inequality first with -pf(vk) and then with pf*(vk), we obtain
(13)-pf(vk)log(pf(vk)) ≥-1ln(2)∑i=1npik·pf(vk) -pf(vk)log(pf*(vk)) ≥-1ln(2)(pf(vk))2-pf(vk)pf*(vk)pf*(vk) -pf(vk)log(pf*(vk)),-pf*(vk)log(pf*(vk)) ≤1ln(2)∑i=1npik·pf*(vk) -pf*(vk)log(pf(vk)) ≤1ln(2)(pf(vk)-pf*(vk)) -pf*(vk)log(pf(vk)),∀1≤k≤|V|.
Finally, summating after k yields the wanted results.

4. Conclusions
As a main result of this paper, we improved two inequalities between entropy-based measures, this giving us a way of bounding measures of network properties. A good example for this purpose is presented into [6], where, using concrete information functionals, the corresponding entropies for the graphs in a set of 2265 nonisomorphic chemical graphs, that is, in MS2265, are computed. Here, new stronger lower and, respectively, upper bounds for entropy-based measures are given and thus, by knowing the limitations of such measures, we narrow down the set of expected values. The utility of inequalities between measures of structural information content is found in problem solving through problem transformation, as is also spotted in [6].