EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6480 ISSN 1307-5543 – ejpam.com Published by New York Business Global Mathai-Haubold Interval Entropy and Related Inequalities Javid Gani Dar1, Sara Mohamed Ahmed Alsheikh2, Prakash Jadhav3,∗ 1 Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed University) (SIU), Lavale, Pune, Maharashtra, India 2 Department of Statistics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia 3 Department of Mechanical Engineering, SRM University AP, Andhra Pradesh, 522240, India Abstract. The paper introduces a generalized interval entropy measure that provides a compre- hensive framework for understanding and quantifying the uncertainty in systems with doubly trun- cated random variables. By characterizing well-known lifetime distributions (exponential, Pareto, and finite range distributions), deriving a lower bound for the entropy, and exploring stochas- tic comparisons, the paper demonstrates the usefulness of this measure in reliability modelling, survival analysis and information theory. 2020 Mathematics Subject Classifications: 62B10, 62R07, 60G35, 62N05 Key Words and Phrases: Entropy, residual entropy, interval entropy, generalized failure rate, characterization 1. Introduction Differential entropy, also referred to as the Shannon information measure [1] is the tra- ditional measure of uncertainty. Shannon outlined the features of information sources and communication channels in order to evaluate the outputs of different sources. In addition to providing a framework for tackling a wide range of statistical problems, statisticians have played a significant role in the development of information theory. The use of infor- mation measures for doubly truncated random variables have been investigated by Sunoj et al. [2] and this work is important for understanding the many components of a sys- tem’s failure between two time points. Most existing studies on information and entropy measures, including Shannon entropy and its variants, have primarily focused on classical settings or extensions such as Rényi and Tsallis entropies. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6480 Email addresses: javid.dar@sitpune.edu.in (J. G. Dar), salshekh@ut.edu.sa (S. M. A. Alsheikh), prakash.j@srmap.edu.in (P. Jadhav) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 2 of 14 Interval entropy is still a relatively new and developing measure in information theory and reliability modeling, so the research gaps mainly lie in its generalizations, applications, and connections to other measures. The associated inequalities, bounds, and related properties have not been systematically developed or studied in depth. In particular, there is a lack of results connecting interval entropy to fundamental mathematical inequalities and their implications for statistical or applied problems. Some Definitions: i) Probability density function: A probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value. For a continuous random variable X, the PDF is denoted by f(x) and has the following properties: a) f(x) ≥ 0 b) P (a ≤ X ≤ b) = ∫ b a f(x)dx c) ∫∞ −∞ f(x)dx = 1 ii) Survival function: For a non-negative random variable X (representing lifetime or time- to-event), the survival function is defined as RX(x) = P(X > x) iii) Hazard rate function: For a non-negative lifetime random variable X, the hazard rate function h(x) is defined as h(x) = lim ∆x→0 P (x < X < x+∆x/X ≥ x) ∆x , x ≥ 0. iv) Mean residual function: It represents the expected remaining lifetime of an item, given that it has already survived up to time t and is given by r(x) = 1 R(t) ∫ ∞ t R(x)dx Also following relation holds: h(x) = f(x) R(x) Let X be a non- negative random variable denoting the life time of a system, a compo- nent or living organism with probability density function f , distribution function F and reliability function R. It is assumed that the component is functioning at t = 0 and it will fail to some t > 0, so that R(0) = 1. The functional relationship between the hazard rate function and mean residual life func- tion is given by h(t) = r′(t) + 1 r(t) (1.1) The average amount of uncertainty associated with the random variable X as given by Shannon entropy, is H(X) = − ∫ ∞ 0 f(x) log f(x)dx (1.2) J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 3 of 14 Mathai- Haubold [3] introduced the generalized information measure Kα(X) = 1 (α− 1) ∫ ∞ 0 ( f(2−α)(x)− 1 ) dx, α 6= 1, 0 < α < 2 (1.3) Ebrahimi [10] argues that (1.1) is no longer effective in evaluating the uncertainty about the remaining life time of the unit if a unit is known to have survived up to an age t. A unit with high uncertainty is less reliable than one with low uncertainty. In order to account for this, he developed a measure of uncertainty for the residual life time distribution known as residual entropy. The residual entropy of a continuous random variable X is defined as K(X; t) = − ∫ ∞ t f(x) R(t) log f(x) R(t) dx (1.4) Corollary: For t = 0, (1.4) reduces to (1.1). Ebrahimi [4] established that a specific dynamic measure can uniquely determine the un- derlying distribution function. This implies that by using this dynamic measure, one can fully understand the distribution of data or a random variable. Belzunce et al. [5], Asadi et al. [6], Dar et al. [7] and Anant et al.[8] extended Ebrahimi’s work, but in the con- text of a generalized residual entropy. This type of entropy is a variation or extension of classical entropy measures and is useful for studying the uncertainty or information content of a system, particularly in cases where there may be a residual or remaining uncertainty after some event. Nair and Rajesh [9] and Asadi and Ebrahimi [10] further explored characterizations of distributions using dynamic entropies. In particular, they might have looked at how various dynamic entropy measures can be used to derive useful properties of distributions that are relevant in engineering, decision-making, or risk assess- ment. Di Crescenzo and M. Longobardi [11, 12] and Nanda, and P. Paul [13] introduces a new measure of uncertainty related to the past lifetime of a system or component, given that failure has already occurred before a certain time. N. Ebrahimi and F. Pellerey [14] introduce a partial ordering of survival functions that reflects the degree of uncertainty or randomness associated with the lifetime of systems. This ordering helps in characterizing and distinguishing different reliability behaviors, such as aging and variability in lifetimes. In many situations, we only have information between two points, so we should study the statistical measures under the condition of doubly truncated random variables. The doubly truncated measures are applicable to engineering systems when the observations are measured after it starts operating and before it fails. If the random variable X de- notes the lifetime of a unit, then the random variable (X/t1 < X < t2); where t1, t2 ∈ D ={ (u, v) ∈ R2 : F (u) < F (v) } is called a doubly truncated lifetime variable. Another exten- sion of Shannon entropy is based on a doubly truncated random variable ( X/t1 < X < t2 ), which is defined as K(X; t1, t2) = − ∫ t2 t1 f(x) R (t1)− R(t2) log f(x) R (t1)− R(t2) dx (1.5) J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 4 of 14 Given that a system has survived up to time t1 and has been found to be down at time t2, then K (X; t1, t2) measures the uncertainty about its lifetimes between t1 and t2. Sunoj et al. [19] have explored the use of information measures for double truncated random variables. Furthermore, Misagh and Yari [15, 16] explored the use of weighted information measures for doubly truncated random variables. For various results on doubly truncated random variable, we refer to Kayal and Moharana [17], Kundu [18], Jalayeri et.al [19], Kumar [20] and in [21, 22]. Joshi and Dar [23] develop MathaiHaubold Fuzzy Entropy extends the MathaiHaubold Entropy framework to the domain of fuzzy set theory, providing a generalized measure of uncertainty and fuzziness in imprecise or vague systems. This formulation integrates the pathway model of Mathai and Haubold with the principles of fuzzy entropy, allowing for a more flexible quantification of uncertainty when dealing with fuzzy membership functions. Oindrali Das, Siddhartha Chakraborty and Biswabrata Pradhan [24] proposed the MathaiHaubold and study its properties. Since generalized entropy plays an important role, in the field of reliability theory and survival analysis, when a system has lifetime between two time points ( t1, t2 ), the concept of doubly truncated data is also applied to medical research, specifically in the study of tumor progression in cancer patients who receive chemotherapy. In these studies, patients’ times to progression (i.e., the time it takes for their condition to worsen) are truncated: they are observed between two points in time, possibly between when they start treatment and when they either progress or die. This doubly truncated data can be modelled effectively using generalized entropy. 2. Mathai-Haubold Interval Entropy Based on the measure defined in (1.3), Dar and Bander [25], introduce the information measure that takes the current age of the system into consideration and generalizes (1.4) as Kα(X; t) = 1 (α− 1) [∫∞ t f2−α(x)dx R2−α(t) − 1 ] , α 6= 1, 0 < α < 2 (2.1) Using equation (2.1),the interval entropy of order α (now onwards MHIE) of the double truncated random variable ( X/t1 < X < t2 ) is given by: Kα (X; t1, t2) = 1 (α− 1) [∫ t2 t1 ( f(x) F (t2)− F (t1) )2−α dx− 1 ] , α 6= 1, 0 < α < 2 (2.2) When the system has age t1, (2.2) provides the information spectrum of the remaining life of the system until age t2. Corollary: For t1 = t and t2 = ∞, (2.2) reduces to (2.1). Definition 2.1: The general failure rate (GFR) of a doubly truncated random variable (X/t1 < X < t2) is defined as hi (t1, t2) = f(ti) F(t2)−F(t1) , i = 1, 2. http://et.al J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 5 of 14 Based on (2.2), we derive the interval entropy of some well-known lifetime distributions: (i) Uniform distribution: The probability density function and corresponding cdf of uniform distribution is given by f(x) = 1 b−a , a < x < b,F(x) = 1− b−x b−a respectively. The MHIE is given by (α− 1)Kα (X; t1, t2) = 1 (t2−t1) 2−α (∫ t2 t1 dx ) − 1 or Kα (X; t1, t2) = 1 (α− 1) [ t2 h2−α 2 (t1, t2)− t1 h2−α 1 (t1, t2)− 1 ] (ii) Exponential distribution: The probability density function and corresponding cdf of exponential distribution is given by f(x) = θe−θx, x > 0, θ > 0, F (x) = 1− e−θx respectively. (α− 1)Kα (X; t1, t2) = ( θ e−θt1−e−θt2 )2−α [∫ t2 t1 e−(2−α)θxdx ] − 1 = 1 θ(2− α) [( θe−θt1 e−θt1−e−θt2 )2−α − ( θe−θt2 e−θt1−e−θt2 )2−α ] − 1 or Kα (X; t1, t2) = 1 (α− 1) [ 1 θ(2− α) ( h2−α 1 (t1, t2)− h2−α 2 (t1, t2) ) − 1 ] (iii) Finite range distribution: The probability density function and corresponding cdf of finite range distribution is given by f(x) = ab(1− ax)b−1, 0 < x < 1 b , a, b > 0, F(x) = 1− (1− ax)b respectively. (α− 1)Kα (X; t1, t2) = ( ab (1− at1) b − (1− at2) b )2−α [∫ t2 t1 (1− ax)(b−1)(2−α)dx ] − 1 Kα (X; t1, t2) = 1 (α− 1)) [ 1 a((2− α)(b− 1) + 1 ( (1− at1) h 2−α 1 (t1, t2)− (1− at2) h2−α 2 (t1, t2) ) − 1 ] (iv) Power distribution: The density function and cdf of power distribution is given as J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 6 of 14 f(x) = b a (x a )b−1 , 0 < x < b, b > 0, F (x) = (x a )b (α− 1)Kα (X; t1, t2) = ( b tb2 − tb1 )2−α [∫ t2 t1 x(b−1)(2−α)dx ] − 1 = 1 ((2− α)(b− 1) + 1) t2( btb−1 2 tb2 − tb1 )2−α − t1 ( btb−1 1 tb2 − tb1 )2−α − 1 or Kα (X; t1, t2) = 1 (α− 1) [ 1 ((2− α)(b− 1) + 1) ( t2 h2−α 2 − t1 h2−α 1 (t1, t2) ) − 1 ] . To gain further insights into the behavior of the Mathai-Haubold Interval Entropy (MHIE), we provide graphical representations of MHIE for several standard lifetime dis- tributions. Figure 1: Uniform Distribution Figure 2: Uniform Distribution Figure 3: Exponential Distribution Figure 4: Exponential Distribution J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 7 of 14 Figure 5: Power Distribution Figure 6: Power Distribution 3. Some characterization Results: Theorem 3.1. Let X be the non-negative random variable having continuous density function f(x) and distribution function F(x). Assume that Kα (X; t1, t2) is increasing with respect to t1 and t2, then Kα (X; t1, t2) uniquely determine the distribution function F(x). Proof: Differentiating (2.2) partially with respect to t1 and t2, we get (α− 1) ∂ ∂t1 Kα (X; t1, t2) = (2− α) ∫ t2 t1 f2−α(x)f (t1) dx ( F (t2)− F (t1)) 3−α − ( f (t1) F (t2)− F (t1) )2−α and (α− 1) ∂ ∂t2 Kα (X; t1, t2) = (2− α) ∫ t2 t1 f2−α(x)f (t1) dx ( F (t2)− F (t1)) 3−α − ( f (t2) F (t2)− F (t1) )2−α After simplification, we get (α− 1) ∂ ∂t1 Kα (X; t1, t2) = −h2−α 1 (t1, t2) + (2− α)(α− 1)h1 (t1, t2)Kα (X; t1, t2) (α− 1) ∂ ∂t2 Kα (X; t1, t2) = −h2−α 2 (t1, t2) + (2− α)(α− 1)h2 (t1, t2)Kα (X; t1, t2) or h2−α 1 (t1, t2) = (2− α)(α− 1)h1 (t1, t2)Kα (X; t1, t2)− (α− 1) ∂ ∂t1 Kα (X; t1, t2) h2−α 2 (t1, t2) = (2− α)(α− 1)h2 (t1, t2)Kα (X; t1, t2) + (α− J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 8 of 14 (i) ∂ ∂t2 Kα (X; t1, t2) Now for any fixed t1 and t2, h1 (t1, t2) and h2 (t1, t2) is a positive solution of the equation ξ (xt2) = 0 and ψ (yt1) = 0, where ξ (xt2) = (xt2) 2−α − (2− α)(α− 1)xt2 Kα (X; t1, t2)− (α− 1) ∂ ∂t2 Kα (X; t1, t2) (3.1) and ψ (yt1) = (yt1) 2−α − (2− α)(α− 1)yt1 Kα (X; t1, t2) + (α− 1) ∂ ∂t1 Kα (X; t1, t2) (3.2) Differentiating partially with respect to xt2 and yt1 , we obtain ξ′ (xt2) = (2− α) (xt2) 1−α − (2− α)(α− 1)Kα (X; t1, t2) (3.3) ψ′ (yt1) = (2− α) (yt1) 1−α − (2− α)(α− 1)Kα (X; t1, t2) (3.4) For extreme values, ξ′ (xt2) = 0, ψ′ (yt1) = 0, which gives xt2 = [(α− 1)Kα (x; t1, t2)] 1 1−α = yt1 . Further more ξ′′ (xt2) = (2− α)(1− α) (xt2) −α and ψ′′ (yt1) = (2− α)(1− α) (yt1) −α Two cases arise: Case I: α < 1, then ξ′′ (xt2) = ψ′′ (yt1) > 0. Thus both ξ (xt2) and ψ (yt1) are minimized at xt2 and yt1 respectively. Also, ξ(0) < 0 and ξ(∞) = ∞. Similarly, ψ(0) > 0, ψ(∞) = ∞. Hence both the equations ξ (xt2) = 0 and ψ (yt1) = 0 have unique positive solutions h1 (t1, t2) and h2 (t1, t2) respectively. Case II: α > 1, then ξ′′ (xt2) = ψ′′ (yt1) < 0. Thus both ξ (xt2) and ψ (yt1) are maximized at xt2 and yt1 respectively. Also, ξ(0) < 0 and ξ(∞) = ∞. Similarly, ψ(0) > 0, ψ(∞) = ∞. Hence both the equations ξ (xt2) = 0 and ψ (yt1) = 0 have unique positive solutions h1 (t1, t2) and h2 (t1, t2) respectively. Combining both the cases, we conclude Kα (X; t1, t2) determines the hi (t1, t2) , i = 1, 2 uniquely. J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 9 of 14 Theorem 3.2. Let X be the non-negative random variable follows uniform distribution over ( a, b ), a < b if and only if Kα (X; t1, t2) = 1 α− 1 [ t2 − t1 (t2 − t1) 2−α − 1 ] (3.5) Proof: The probability density function and corresponding cdf of uniform distribution is given by f(x) = 1 b− a , a < x < b, F(x) = 1− b− x b− a respectively. Using (2.2), we get Kα (X; t1, t2) = 1 α− 1 [ t2 − t1 (t2 − t1) 2−α − 1 ] , which proves only if part of the theorem. To prove the if part, let (3.5) is valid. Thus from (2.2) and (3.5), we get∫ t2 t1 f2−α(x)dx = (t2 − t1) 2−α( (F (t2)− F (t1)) α−2 Differentiating with respect t1 and t2, we obtain h2−α 1 (t1, t2) = (α− 1) (t2 − t1) α−2 + (2− α) (t2 − t1) α−1 h1 (t1, t2) and h2−α 2 (t1, t2) = (α− 1) (t2 − t1) α−2 + (2− α) (t2 − t1) α−1 h2 (t1, t2) Now for any fixed t1 and t2, h1 (t1, t2) and h2 (t1, t2) is a positive solution of the equation ξ (xt2) = 0 and ψ (yt1) = 0, where ξ (xt2) = (xt2) 2−α − (α− 1) (t2 − t1) 2−α − (2− α) (t2 − t1) 1−αxt2 (3.6) and ψ (yt1) = (yt1) 2−α − (α− 1) (t2 − t1) 2−α − (2− α) (t2 − t1) 1−αyt1 (3.7) Differentiating both side of (3.6) and (3.7) with respect to xt2 and yt1 respectively, we get ξ′ (xt2) = (2− α) (xt2) 1−α − (2− α) (t2 − t1) 1−α ψ′ (yt1) = (2− α) (yt1) 1−α − (2− α) (t2 − t1) 1−α J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 10 of 14 For extreme values, ξ′ (xt2) = 0, ψ′ (yt1) = 0, which gives xt2 = 1 t2 − t1 = h1 (t1, t2) and yt1 = 1 t2 − t1 = h2 (t1, t2) , which proves the theorem. Theorem 3.3. Let X be the non-negative random variable having continuous density function f(x) and distribution function F (x), then a relation of the form (α− 1)Kα (X; t1, t2) = 1 c [ (1 + kt2) h 2−α 2 (t1, t2)− (1 + kt1) h 2−α 1 (t1, t2) ] − 1 (3.8) where c is a constant holds for all t1, t2. Then X has (i) An Exponential distribution iff c = 0 (ii) A Pareto distribution iff c < 0 (iii) A finite range distribution iff c > 0. Proof. (i) The probability density function and corresponding cdf of exponential dis- tribution is given by f(x) = θe−θx, x > 0, θ > 0, F (x) = 1− e−θx respectively Also, hi (t1, t2) = θe−θti e−θt1−e−θt2 Using (2.2), we get (α− 1)Kα (X; t1, t2) = 1 c [ h2−α 2 (t1, t2)− h2−α 1 (t1, t2) ] − 1 (3.9) where c = θ(2− α). Thus (3.9) is equivalent to (3.8) for k = 0. (ii) The probability density function and corresponding cdf of uniform distribution is given by f(x) = rs(1 + rx)−(s+1), r, s > 0, FX(x) = 1− (1 + rx)s Thus using (2.2), we have (α− 1)Kα (X; t1, t2) = 1 c [ (1 + rt2) h 2−α 2 (t1, t2)− (1 + rt1) h 2−α 1 (t1, t2) ] − 1 where c = 1 r[1−(s+1)(2−α)] and k = r > 0. Thus (3.8) holds (iii) The probability density function and corresponding cdf of uniform distribution is given by J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 11 of 14 f(x) = ab(1− ax)b−1, 0 < x < 1 b , a, b > 0, F(x) = 1− (1− ax)b respectively. Using (2.2), one will get (α− 1)Kα (X; t1, t2) = 1 c [ (1− at2) h 2−α 1 (t1, t2)− (1− at1) h 2−α 2 (t1, t2) ] − 1 where c = 1 −a[1+(b−1)(2−α)] and k = −a < 0. Thus (3.8) holds. Conversely, let (3.8) holds. Using (2.2) in (3.8), we get c ∫ t2 t1 f2−α(x)dx = (1 + kt2) 2−α f2−α (t2)− (1 + kt1) 2−α f2−α (t1)− 1. Differentiating with respect to t2, keeping t1 fixed f′(t2) f(t2) = ( c−k 2−α )( 1 1+kt2 ) . Similarly differentiating with respect to t1, keeping t2 fixed f′(t1) f(t1) = ( c−k 2−α )( 1 1+kt1 ) . Generally, f′(ti) f(ti) = ( c−k 2−α )( 1 1+kti ) , i = 1, 2. This gives d dt log f (ti) = ( c− k 2− α )( 1 1 + kti ) , i = 1, 2 (3.10) (3.10) clearly shows that the underlying distribution is exponential if k = 0, Pareto distribution for k > 0 and finite rage distribution for k < 0. Hence the theorem is proved. 4. Some Inequalities and stochastic comparison: Theorem 4.1. Let X be an absolutely continuous random variable density function f(x) and distribution function F(x), then for α > 1(α < 1), then Kα (X; t1, t2) is increasing (decreasing) in t2 if F (t1) = 0 Proof: Kα (X; t1, t2) = 1 (α−1) [∫ t2 t1 ( f(x) F(t2)−F(t1) )2−α dx− 1 ] For F (t1) = 0, we have Kα (X; t1, t2) = 1 (α−1) [∫ t2 t1 ( f(x) F(t2) )2−α dx− 1 ] Differentiating with respect t2, we get K′ α (X; t1, t2) = F(t2) F3−α(t2) [ f1−α (t2) F (t2)− (2− α) ∫ t2 t1 f2−α(x)dx ] Define K(t2) = f1−α (t2) F (t2)− (2− α) ∫ t2 t1 f2−α(x)dx Differentiating again with respect t2, we have K′ (t2) = (α− 1)f−α (t2) ( f2 (t2)− F (t2) f ′ (t2) . If α > 1, then K′ (t2) ≥ 0, which in turns gives that Kα (X; t1, t2) is an increasing function of t2. If α < 1, then K′ (t2) ≤ 0, which in turns gives that Kα (X; t1, t2) is an decreasing function of t2. J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 12 of 14 Theorem 4.2. For an absolutely continuous random variable X , if Kα (X; t1, t2) is in- creasing (decreasing) in t1 for fixed t2, then h1 (t1, t2) ≤ (≥) [((2− α)(α− 1)Kα (X; t1, t2))] 1 1−α for α > 1 h1 (t1, t2) ≥ (≤) [((2− α)(α− 1)Kα (X; t1, t2))] 1 1−α for α < 1. Proof: We Know that (α− 1) ∂ ∂t1 Kα (X; t1, t2) = −h2−α 1 (t1, t2) + (2− α)(α− 1)h1 (t1, t2)Kα (X; t1, t2) if Kα (X; t1, t2) is increasing (decreasing) in t1 for fixed t2, then −h2−α 1 (t1, t2) + (2− α)(α− 1)h1 (t1, t2)Kα (X; t1, t2) ≥ (≤)0 h2−α 1 (t1, t2) ≤ (≥)(2− α)(α− 1)h1 (t1, t2)Kα (X; t1, t2) h1−α 1 (t1, t2) ≤ (≥)(2− α)(α− 1)Kα (X; t1, t2) For α > 1 h1 (t1, t2) ≤ (≥) [((2− α)(α− 1)Kα (X; t1, t2))] 1 1−α For α < 1 h1 (t1, t2) ≥ (≤) [((2− α)(α− 1)Kα (X; t1, t2))] 1 1−α Theorem 4.3. For an absolutely continuous random variable X , if Kα (X; t1, t2) is in- creasing (decreasing) in t2 for fixed t1, then h2 (t1, t2) ≤ (≥) [((2− α)(α− 1)Kα (X; t1, t2))] 1 1−α for α > 1 h2 (t1, t2) ≥ (≤) [((2− α)(α− 1)Kα (X; t1, t2))] 1 1−α for α < 1. Proof: The proof follows under similar lines as in theorem 4.2. Let X and Y be two random variables. Also, the distribution function and density function of X are indicated by F (x) and f(x) and those of Y are denoted by G(x) and g(x) respectively, then X is said to be less than or equal to Y in likelihood ratio ordering, if F (x) ≥ G(x), x ≥ 0. We write X ≤st Y . Theorem 4.4. Let X and Y be an absolutely continuous random variable density function with cdf F(x) and G(x) respectively. If X ≤st Y for all t1, t2 ≥ 0, then and distribution function F(x), then for α > 1(0 < α < 1), then Kα (X; t1, t2) ≥ (≤)Kα (Y; t1, t2) for α > 1 and for Proof: Since X ≤st Y , therefore 1 (α− 1) [∫ t2 t1 ( f(x) F (t2)− F (t1) )2−α dx− 1 ] ≥ 1 (α− 1) [∫ t2 t1 ( g(x) G (t2)−G (t1) )2−α dx− 1 ] Thus for α > 1, we have Kα (X; t1, t2) ≥ Kα (Y; t1, t2) and for 0 < α < 1, it is obvious Kα (X; t1, t2) ≤ Kα (Y; t1, t2) J. G. Dar, S. M. A. Alsheikh, P. Jadhav / Eur. J. Pure Appl. Math, 18 (4) (2025), 6480 13 of 14 5. Conclusion: Generalized interval entropy plays a critical role in information theory, reliability mod- eling and survival analysis by quantifying the uncertainty associated with a system’s life- time, particularly when it has already survived between two time points. The measure is useful in characterizing different lifetime distributions and establishing relationships with reliability measures such as survival functions and hazard rates. Additionally, the con- nection between generalized interval entropy and stochastic ordering provides a powerful tool for comparing different lifetime distributions. Overall, the application of generalized interval entropy in the context of doubly truncated random variables and its generalization of previous results represents an important advancement in understanding and modeling the uncertainty inherent in lifetime distributions. 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