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Introduction: Accurate placement of an endotracheal tube (ETT) is critical for patient safety during medical procedures. The research explores the potential of ultrasound technology to provide a reliable alternative for ETT confirmation, offering insights into its diagnostic performance and the implications for enhancing patient care in surgical settings. Aims & Objectives: To determine the diagnostic accuracy of ultrasound for the confirmation of endotracheal tube keeping Capnography as a Gold standard. Place and Duration of Study: It is a Cross Sectional study and the study was carried out in Operation Theater Shaikh Zayed Hospital Lahore, within 6 months after approval of synopsis i.e. from 5th May, 2020 till 4th November, 2021. Material & Methods:Total 219 patients who fulfilled the inclusion criteria were enrolled. After standardization of anesthetic measures all patients were intubated by direct laryngoscopy. ETT placement was assessed by capnometry and by ultrasonography. Endotracheal tube placement was labeled (as per operational definition) on both the techniques.The analysis of the data was conducted using SPSS version 21.0, a p-value of ?0.05 was considered significant. Results: The mean age, BMI, neck circumference and thyromental distance of the patients was 39±8.15 years, 27±3.42 Kg/m 2, 29.5±4.63cm and 6.9±0.50cm. There were 58% males and 42% females in the study. Ultrasonography exhibited sensitivity, specificity, positive predictive value, negative predictive value, and diagnostic accuracy rates of 98.5%, 90.6%, 98.9%, 86.1%, and 97.7%, respectively, in its ability to detect the accurate placement of endotracheal tubes (ETT). Conclusion: The sensitivity, specificity, diagnostic accuracy and promptness of recognition for confirmation of tracheal placement of ETT is higher withultrasonography compared to the gold standardcapnography in patients undergoing elective surgery under general anesthesia.

Objective: To determine the frequency of need for postoperative intravenous analgesia after rectal diclofenac suppository in patients undergoing lower abdominal gynecological or obstetrical surgery under general anesthesia Study Design: Cross-sectional study. Place and Duration of Study: The study was carried out at the Operation Theater and Postoperative Ward of Sheikh Zayed Hospital, Lahore, Pakistan from January 2021 to July 2021. Methods: A total of 245 women undergoing gynecological and obstetrical surgeries under general anesthesia were enrolled in the study using non-probability consecutive sampling. All the participants were recruited after obtaining informed consent and meeting the inclusion criteria. All subjects had a single 100mg diclofenac suppository inserted rectally after undergoing general anesthesia, and their Visual Analog Scale (VAS) scores for postoperative analgesia were recorded immediately after surgery and again at 1, 6, and 24 hours. Results: The mean age of the patients was 37 years, with a standard deviation of 6.2. The mean VAS (Visual Analog Scale) scores at 1, 6, 12, and 24 hours were 3 (±1.24), 5 (±1.46), 7 (±1.93), and 6 (±2.14), respectively. The need for rescue analgesia was reported in 15.9% of cases at 1 hour, 37.1% at 6 hours, 82% at 12 hours, and 68.2% at 24 hours. Conclusion: The administration of diclofenac via rectal placement led to a notable decrease in Visual Analog Scale scores and a lower incidence of needing additional pain relief within the first 6 hours after surgery. How to cite this: Saeed R, Mahboob A, Ali SM, Latif A, Ammar T, Kakepotto IA. Requirement of Postoperative Analgesia after Per Rectal Placement of Diclofenac Sodium in Patients Undergoing Gynecological and Obstetrical Surgeries under General Anesthesia. Life and Science. 2024; 5(1): 120-125. doi: http://doi.org/10.37185/LnS.1.1.515

In this study, some mappings related to the Fejér-type inequalities for harmonically convex functions are defined over [0,1]. Some Fejér-type inequalities for harmonically convex functions are proved using these mappings. Properties of these mappings are considered and consequently, refinements are obtained of some known results.

The purpose of this article is to discuss some midpoint type HHF fractional integral inequalities and related results for a class of fractional operators (weighted fractional operators) that refer to harmonic convex functions with respect to an increasing function that contains a positive weighted symmetric function with respect to the harmonic mean of the endpoints of the interval. It can be concluded from all derived inequalities that our study generalizes a large number of well-known inequalities involving both classical and Riemann-Liouville fractional integral inequalities.

In accordance with the quantum calculus, the quantum Hermite-Hadamard type inequalities shown in recent findings provide improvements to quantum Hermite-Hadamard type inequalities. We acquire a new $ q{_{\kappa_1}} $-integral and $ q{^{\kappa_2}} $-integral identities, then employing these identities, we establish new quantum Hermite-Hadamard $ q{_{\kappa_1}} $-integral and $ q{^{\kappa_2}} $-integral type inequalities through generalized higher-order strongly preinvex and quasi-preinvex functions. The claim of our study has been graphically supported, and some special cases are provided as well. Finally, we present a comprehensive application of the newly obtained key results. Our outcomes from these new generalizations can be applied to evaluate several mathematical problems relating to applications in the real world. These new results are significant for improving integrated symmetrical function approximations or functions of some symmetry degree.

Abstract In this paper, we obtain new Hermite–Hadamard-type inequalities for r-convex and geometrically convex functions and, additionally, some new Hermite–Hadamard-type inequalities by using the Hölder–İşcan integral inequality and an improved power-mean inequality.

In this study, the assumption of being differentiable for the convex function f in the (p, q)-Hermite-Hadamard inequality is removed. A new identity for the right-hand part of (p, q)-Hermite-Hadamard inequality is proved. By using established identity, some (p, q)-trapezoid integral inequalities for convex and quasi-convex functions are obtained. The presented results in this work extend some results from the earlier research.

In this paper, we present some mappings defined over 0,1 related to the Fejér-type inequalities that have been established for harmonically convex functions. As a consequence, we obtain companions of Fejér-type inequalities for harmonically convex functions by using these mappings. Properties of these mappings are discussed, and consequently, we obtain refinement inequalities of some known results.

In this paper we consider a fractional nonlinearity for the wave equation with friction and viscoelastic damping. Using Fixed point theorem a global in time existence of small data solutions to the Cauchy problem is investigated in this research. Our main interest is to show the influence of the fractional nonlinearity parameter to the admissible range of exponent ς comparing with power nonlinearity and also the generating of loss of decay.

In this study, on the subject of harmonic convex functions, we introduce some new functionals linked with weighted integral inequalities for harmonic convex functions. In addition, certain new inequalities of the Fejér type are discovered.

Throughout this study, the concept of symmetrized harmonically convex stochastic processes will be discussed in further detail. Some certain characterizations for symmetrized harmonically convex stochastic processes are discussed that use Hermite–Hadamard-type inequalities. A Hyers–Ulam-type stability result for harmonically convex stochastic processes is given as well.

Abstract In the article, we present several Hermite–Hadamard type inequalities for the co-ordinated convex and quasi-convex functions and give an application to the product of the moment of two continuous and independent random variables. Our results are generalizations of some earlier results. Additionally, an illustrative example on the probability distribution is given to support our results.

The purpose of this paper is to establish weighted version of Ostrowski type integral inequalities. The inequalities are obtained by using a newly developed special type of five steps weighted kernel. The introduction of this new Kernel gives some new error bounds for various quadrature rules. Applications for Cumulative Distributive Functions are considered.

The objective of this manuscript is to establish a link between the concept of inequalities and Center-Radius order functions, which are intriguing due to their properties and widespread use. We introduce the notion of the CR (Center-Radius)-order interval-valued preinvex function with the help of a total order relation between two intervals. Furthermore, we discuss some properties of this new class of preinvexity and show that the new concept unifies several known concepts in the literature and also gives rise to some new definitions. By applying these new definitions, we have amassed many classical and novel special cases that serve as applications of the key findings of the manuscript. The computations of cr-order intervals depend upon the following concept B=⟨Bc,Br⟩=⟨B¯+B̲2,B¯−B̲2⟩. Then, for the first time, inequalities such as Hermite–Hadamard, Pachpatte, and Fejér type are established for CR-order in association with the concept of interval-valued preinvexity. Some numerical examples are given to validate the main results. The results confirm that this new concept is very useful in connection with various inequalities. A fractional version of the Hermite–Hadamard inequality is also established to show how the presented results can be connected to fractional calculus in future developments. Our presented results will motivate further research on inequalities for fractional interval-valued functions, fuzzy interval-valued functions, and their associated optimization problems.

Fractional integrals and inequalities have recently become quite popular and have been the prime consideration for many studies. The results of many different types of inequalities have been studied by launching innovative analytical techniques and applications. These Hermite–Hadamard inequalities are discovered in this study using Atangana–Baleanu integral operators, which provide both practical and powerful results. In this paper, a symmetric study of integral inequalities of Hermite–Hadamard type is provided based on an identity proved for Atangana–Baleanu integral operators and using functions whose absolute value of the second derivative is harmonic convex. The proven Hermite–Hadamard-type inequalities have been observed to be valid for a choice of any harmonic convex function with the help of examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains.

The main purpose of this research is to concentrate on the development of new definitions for the weighted geometric fractional integrals of the left-hand side and right-hand side of the function ℵ with regard to an increasing function used as an integral kernel. Moreover, the newly developed class of left-hand side and right-hand side weighted geometric fractional integrals of a function ℵ, by applying an additional increasing function, identifies a variety of novel classes as special cases. This is a development of the previously established fractional integrals by making use of the class of geometrically convex functions. Geometrically convex functions in weighted fractional integrals of a function ℵ in the form of another rising function yield the Hermite–Hadamard inequality type. We also establish a novel midpoint identity and the associated inequalities for a class of weighted fractional integral functions known as geometrically convex with respect to an increasing function and symmetric with respect to the geometric mean of the endpoints of the interval. In order to demonstrate the validity of our research, we present examples. Moreover, fractional inequalities and their solutions are applied in many symmetrical domains.

In this paper, new weighted Hermite–Hadamard type inequalities for differentiable h-convex and quasi h-convex functions are proved. These results generalize many results proved in earlier works for these classes of functions. Applications of some of our results to s˘-divergence and to statistics are given.

In this paper, we present some new refinement mappings associated with the Hermite–Hadamard type inequalities that are constructed for GA-convex mappings. Our investigation of the mappings leads to the discovery of several interesting features as well as the development of some inequalities for the Hermite–Hadamard type inequalities, which have already been established for GA-convex functions, as well as refining the relationship between the middle, rightmost, and leftmost elements of the function. Some applications to special means of positive real numbers are also given.

In this article, we present several new inequalities involving the κ-fractional integral for the integrable function ℱ which satisfies one of the following conditions: aℱq is preinvex for some q>1; bℱ′ is bounded; cℱ′ is a Lipschitz function. As applications, we establish new inequalities for the weighted arithmetic and generalized logarithmic means.

Abstract This paper presents new weighted Hermite–Hadamard type inequalities for a new class of convex functions which are known as geometrically quasi-convex functions. Some applications of these results to special means of positive real numbers have also been presented. These findings have been proved to be useful for researchers working in the fields of numerical analysis and mathematical inequalities.

In this paper, we establish a new version of Hermite-Hadamard-Fejér type inequality for harmonically convex functions in the form of weighted fractional integral. Secondly, an integral identity and some weighted midpoint fractional Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions by involving a positive weighted symmetric functions have been obtained. As shown, all of the resulting inequalities generalize several well-known inequalities, including classical and Riemann–Liouville fractional integral inequalities.

In this article, firstly, we establish a novel definition of weighted interval-valued fractional integrals of a function Υ˘ using an another function ϑ(ζ˙). As an additional observation, it is noted that the new class of weighted interval-valued fractional integrals of a function Υ˘ by employing an additional function ϑ(ζ˙) characterizes a variety of new classes as special cases, which is a generalization of the previous class. Secondly, we prove a new version of the Hermite-Hadamard-Fejér type inequality for h-convex interval-valued functions using weighted interval-valued fractional integrals of a function Υ˘ according to another function ϑ(ζ˙). Finally, by using weighted interval-valued fractional integrals of a function Υ˘ according to another function ϑ(ζ˙), we are establishing a new Hermite-Hadamard-Fejér type inequality for harmonically h-convex interval-valued functions that is not previously known. Moreover, some examples are provided to demonstrate our results.

In this paper, several applications of the Hermite–Hadamard inequality for fractional integrals using G A -convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Gamma and incomplete Gamma functions. [ABSTRACT FROM AUTHOR]

In this paper, we establish new (p,q)κ1-integral and (p,q)κ2-integral identities. By employing these new identities, we establish new (p,q)κ1 and (p,q)κ2- trapezoidal integral-type inequalities through strongly convex and quasi-convex functions. Finally, some examples are given to illustrate the investigated results.

In this paper, some new inequalities Hermite-Hadamard type are obtained for functions whose nth derivatives in absolute value are s-convex functions. From our results, several inequalities of Hermite-Hadamard type can be derived in terms of functions whose first and second derivatives in absolute value are s-convex functions as special cases. Our results may provide refinements of some results already exist in literature. Applications to trapezoidal rule and to special means of established results are given.

In this paper some new Hadamard-type inequalities for functions whose derivatives in absolute values are convex are established. Some applications to special means of real numbers are given. Finally, we also give some applications of our obtained results to get new error bounds for the sum of the midpoint and trapezoidal formulae.

In this paper, new Fejér-type inequalities for harmonically convex functions are obtained. Some mappings related to the Fejér-type inequalities for harmonically convex are defined. Properties of these mappings are discussed and, as a consequence, we obtain refinements of some known results. [ABSTRACT FROM AUTHOR]

In this study, the Hermite–Hadamard–Fejér inequalities for G A -h-convex are proved, and the results for particular classes of functions are highlighted. In addition, several generalizations of the Hermite–Hadamard inequalities are presented. Some features of functions H and F that are naturally linked to the Hermite–Hadamard–Fejér-type inequalities for G A -h-convex have also been discussed. Finally, we obtain applications of the results related to the p-logarithmic mean and the mean of order p. [ABSTRACT FROM AUTHOR]

In the present paper, we aim to prove a new lemma and quantum Simpson’s type inequalities for functions of two variables having convexity on co-ordinates over [ α , β ] × [ ψ , ϕ ] . Moreover, our deduction introduce new direction as well as validate the previous results.

Two-point fuzzy Ostrowski type inequalities are proved for fuzzy Hölder and fuzzy differentiable functions. The two-point fuzzy Ostrowski type inequality for M-lipshitzian mappings is also obtained. It is proved that only the two-point fuzzy Ostrowski type inequality for M-lipshitzian mappings is sharp and as a consequence generalize the two-point fuzzy Ostrowski type inequalities obtained for fuzzy differentiable functions.

In this paper, some estimates of third and fourth inequalities in Hermite–Hadamard-type inequalities for coordinated convex functions are proved using the non-additivity of the integrals and Fubini's theorem for fuzzy integrals. That is, the results are obtained in the fuzzy context and using the Lebesgue measure. Several examples are provided on how to evaluate these estimates in order to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new version of Hermite-Hadamard-Fejér type inequality for harmonically convex functions in the form of weighted fractional integral. Secondly, an integral identity and some weighted midpoint fractional Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions by involving a positive weighted symmetric functions have been obtained. As shown, all of the resulting inequalities generalize several well-known inequalities, including classical and Riemann–Liouville fractional integral inequalities. [ABSTRACT FROM AUTHOR]

In this study, we introduce some new mappings in connection with Hermite-Hadamard and Fejer type integral inequalities which have been proved using the GA-convex functions. As a consequence, we obtain certain new inequalities of the Fejer type that provide refinements of the Hermite-Hadamard and Fejer type integral inequalities that have already been obtained.

In this study, trapezoidal-type inequalities in fuzzy settings have been investigated. The theory of fuzzy analysis has been discussed in detail. The integration by parts formula of analysis of fuzzy mathematics has been employed to establish an equality. Trapezoidal-type inequality for functions with values in the fuzzy number-valued space is proven by applying the proven equality together with the properties of a metric defined on the set of fuzzy number-valued space and Höler’s inequality. The results proved in this research provide generalizations of the results from earlier existing results in the field of mathematical inequalities. An example is designed by defining a function that has values in fuzzy number-valued space and validated the results numerically using the software Mathematica (latest v. 14.1). The p-levels of the defined fuzzy number-valued mapping have been shown graphically for different values of p∈0,1.

In this paper, we offer Ostrowski-type inequalities that extend the findings that have been proven for functions of one variable with values in Banach spaces, conducted in a remarkable study by Dragomir, to functions of two variables containing values in the product Banach spaces. Our findings are also an extension of several previous findings that have been established for functions of two variable functions. Prior studies on Ostrowski-type inequalities incriminated functions that have values in Banach spaces or Hilbert spaces. This study is unique and significant in the field of mathematical inequalities, and specifically in the study of Ostrowski-type inequalities, because they have been established for functions having values in a product of two Banach spaces.

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative.

In this paper, using positive symmetric functions, we offer two new important identities of fractional integral form for convex and harmonically convex functions. We then prove new variants of the Hermite-Hadamard-Fejér type inequalities for convex as well as harmonically convex functions via fractional integrals involving an exponential kernel. Moreover, we also present improved versions of midpoint type Hermite-Hadamard inequality. Graphical representations are given to validate the accuracy of the main results. Finally, applications associated with matrices, q-digamma functions and modifed Bessel functions are also discussed.

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives.

In this study, Ostrowski-type inequalities in fuzzy settings were investigated. A detailed theory of fuzzy analysis is provided and utilized to establish the Ostrowski-type inequality in the fuzzy number-valued space. The results obtained in this research not only provide a generalization of the results of Dragomir but also give an extended version of the Ostrowski-type inequalities obtained by Anastassiou.

In this paper, several applications of the Hermite–Hadamard inequality for fractional integrals using GA-convexity are discussed, including some new refinements and similar extensions, as well as several applications in the Gamma and incomplete Gamma functions.

In this paper, new Fejér-type inequalities for harmonically convex functions are obtained. Some mappings related to the Fejér-type inequalities for harmonically convex are defined. Properties of these mappings are discussed and, as a consequence, we obtain refinements of some known results.

In this study, the Hermite–Hadamard–Fejér inequalities for GA-h-convex are proved, and the results for particular classes of functions are highlighted. In addition, several generalizations of the Hermite–Hadamard inequalities are presented. Some features of functions H and F that are naturally linked to the Hermite–Hadamard–Fejér-type inequalities for GA-h-convex have also been discussed. Finally, we obtain applications of the results related to the p-logarithmic mean and the mean of order p.

In this paper, some estimates of third and fourth inequalities in Hermite–Hadamard-type inequalities for coordinated convex functions are proved using the non-additivity of the integrals and Fubini’s theorem for fuzzy integrals. That is, the results are obtained in the fuzzy context and using the Lebesgue measure. Several examples are provided on how to evaluate these estimates in order to illustrate the obtained results.

In this paper, we prove the Hermite–Hadamard–Fejér type inequalities for coordinated h1,h2-convex functions on the rectangle from the plane R2. Some generalizations of the Hermite–Hadamard-type inequalities of two variables are also obtained as a consequence. Some properties of two functionals which are connected with the coordinated h1,h2-convex functions are provided as well. Finally, we give applications of the acquired results to special means of positive real numbers.

In this paper, we present some new and novel mappings defined over 0,1 with the help of GA-convex functions. As a consequence, we obtain companions of Fejér-type inequalities for GA-convex functions with the help of these mappings, which provide refinements of some known results. The properties of these mappings are discussed as well.