Abstract. In this paper we introduce the notion of ϕ -convex functions as generalization of convex functions. Some basic results under various conditions for the function ϕ are investigated. Moreover, we establish Jensen and Hermite-Hadamard type inequalities related to ϕ -convex functions. Also, the notions of ϕ b -convex and ϕ E -convex functions, which are generalization of ϕ -convex functions are introduced and some new results related to these new settings are obtained.Mathematics subject classification (2010): 26A51, 26D15, 52A01.

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Parkinson's disease (PD) is a frequent progressive neurodegenerative disorder. Impaired mitochondrial function is a major feature of sporadic PD. Some susceptibility or causative genes detected in PD are strongly associated with mitochondrial dysfunction including PGC1α, TFAM and GSK3β. microRNAs (miRNAs) are non‐coding RNAs whose altered levels are proven in disparate PD models and human brains. Therefore, the aim of this study was to detect modulations of miRs upstream of PGC1α, TFAM and GSK3β in association with PD onset and progress. In this study, a total of 33 PD subjects and 25 healthy volunteers were recruited. Candidate miRNA (miR‐376a) was selected through target prediction tools and literature survey. Chronic and acute in vitro PD models were created by MPP+‐intoxicated SHSY5Y cells. The levels of miR‐376a and aforementioned genes were assessed by RT‐qPCR. The expression of target genes was decreased in chronic model while there were dramatically up‐regulated levels of those genes in acute model of PD. miR‐376a was strongly altered in both acute and chronic PD models as well as PBMCs of PD patients. Our results also showed overexpression of PGC1α, and TFAM in PBMCs is inversely correlated with down‐regulation of miR‐376a, suggesting that miR‐376a possibly has an impact on PD pathogenesis through regulation of these genes which are involved in mitochondrial function. miR‐376a expression in PD‐derived PBMCs was also correlated with disease severity and may serve as a potential biomarker for PD diagnosis. This is the first study showing altered levels of miR‐376a in PD models and PBMCs, suggesting the probable role of this miRNA in PD pathogenesis. The present study also proposed TFAM and PGC1α as target genes of miR‐376a for the first time, through which it possibly can exert its impact on PD pathogenesis.