The aim of this article is to define a special case of h-convex function, namely the notion of a trigonometrically convex function. Using the Hölder, Hölder-İşcan integral inequality and the power-mean, improved power-mean integral inequalities, and together with an integral identity, some new Simpson-type inequalities have been obtained for trigonometric convex functions. We also give some applications for special means.

In this paper, a new class of convex functions, which is called exponential trigonometric convex functions is studied. We obtain some refinements of the Hermite-Hadamard type inequalities via functions whose second derivatives in absolute value at certain power are exponential trigonometric convex functions. The results of the research by the means of Hölder-İşcan and improved-power-mean integral inequalities.

Background: In dentistry, needles are the most feared and anxiety-causing tool, making anesthetic injection a worrying practice for patients. Aim: To evaluate the effect of intraosseous anesthesia (IOA) and needle-free dental anesthesia (NFA) on pain levels in systemically healthy 8–10-year-old patients. Patients and Methods: Twenty patients aged 8–10 years were included in this cross-over study. Specifically, the pain was measured by Wong Baker, pulse rate (PR), and salivary opiorphin levels (SOL). In addition, the Frankl Behavioral scale was used to measure behaviors and face, legs, activity, cry, consolability (FLACC) was utilized to measure pain and discomfort. To determine the patients' anxiety levels Spielberger State-Trait Anxiety Scale (SSAS-SAAS) was used. The Friedman and Wilcoxon signed-rank tests were used. P < 0.05 was considered significant. Results: According to FLACC scores, IOA and NFA exhibited significantly pain alteration patterns in during local and topical anesthesia, respectively (p = 0.004, 0.001; P < 0.01). Also, only NFA showed significantly decreased SOL values in 5- and 10-min after local anesthesia periods compared to the before levels (p = 0.004, P = 0.001; P < 0.01). Conclusion: Patients feel similar pain perceptions during local anesthesia application in both injection systems. According to the SOL values, NFA may provide more higher anesthetic efficiency than IOA.

In the paper, we introduce the class of trigonometrically convex functions and using the Hölder, Hölder-İşcan, Power-mean and Improved power-mean integral inequality together with an identity we establish some new inequalities of Ostrowski-type for functions whose second derivatives are trigonometrically convex which is a special case of h-convex functions. Some applications for special means are also given.