By using the way of weight functions and Hadamard's inequality, a half-discrete Hilbert-type inequality similar to Mulholland's inequality with a best constant factor is given. The extension with multi-parameters, the equivalent forms as well as the operator expressions are also considered. MSC: 26D15

By using the weight functions and the idea of introducing parameters, a half-discrete Hilbert inequality with a nonhomogeneous kernel and its equivalent form are given. The equivalent statements of the constant factor are best possible related to parameters, and some particular cases are considered. The cases of a homogeneous kernel are also deduced.

By the use of the weight coefficients, the idea of introducing parameters and the Euler-Maclaurin summation formula, a reverse extended Hardy-Hilbert inequality and the equivalent forms are given. The equivalent statements of the best possible constant factor related to a few parameters and some particular cases are also considered. MSC: 26D15

By applying the method of weight functions and the technique of real analysis, a multidimensional Hilbert-type integral inequality with multi-parameters and the best possible constant factor related to the gamma function is given. The equivalent forms and the reverses are obtained. We also consider the operator expressions and a few particular results related to the kernels of non-homogeneous and homogeneous. MSC: 26D15; 47A07; 37A10

Objective: To observe the efficacy and safety of post-operative long-term low-dose oral administration of clarithromycin in patients with refractory chronic rhinosinusitis (RCRS). Explore the characteristics of post-operative microbiota in the nasal cavity in patients with RCRS and compare the differences and changes in microbiota in the nasal cavity before and after medication.Method: This was a prospective, self-controlled study. 18 RCRS patients who had persistent symptoms after endoscopic sinus surgery (ESS) and standard therapy with normal IgE and eosinophil level were included. Low dose (250 mg, QD) clarithromycin was orally administrated for 12 weeks, Symptom severity and endoscopic findings were evaluated before, after 4-week and 12-week of treatment, and nasal cavity microbiota was analyzed simultaneously.Results: A total of 18 RCRS patients were enrolled, and 17 patients completed the study. Four weeks after oral administration of clarithromycin, significant improvement was observed in subjective symptoms including nasal congestion, rhinorrhoea, postnasal drip and general discomfort, as well as endoscopic findings including general surgical cavity condition, rhinedema and rhinorrhoea (P<0.05). After continuous treatment to the 12th week, the improvements of symptoms showed significant improvement compared with baseline, and endoscopic score showed significant improvement compared with both baseline and 4 weeks after treatment. Analysis of middle nasal meatus flora revealed a significant decrease of Streptococcus pneumoniae after 12 weeks of clarithromycin treatment (P<0.05), while the richness, composition, and diversity similar before and after treatment. Patients enrolled experienced no adverse drug reaction or allergic reaction, nor clinically significant liver function impairment observed. Conclusion:Post-operative low-dose long-term oral administration of clarithromycin in RCRS patients can improve the clinical symptoms and facilitate the mucosal epithelization, with good tolerance and safety. The efficacy of clarithromycin in RCRS patients may be related to its regulatory effect on nasal cavity microbiota.