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Thin-layer drying kinetics of pistachio nuts were examined experimentally as a function of drying conditions in a fluidized bed dryer with microwave pretreatment. Four drying specifications of diffusivity, shrinkage, specific energy consumption and total color change were calculated and the effects of parameters were studied. Numerous experimentations were conducted at three levels of air temperature (40, 55, 70 °C), air velocity (1.2, 2.93, 4.01 m/s), and microwave power (270, 450, 630 W). The variation ranges of diffusivity, shrinkage, energy consumption and color change were recorded from 5.01×10–10 to 5.07×10–9 m2/s, from 26.95 % to 13.13 %, from 1.04 to 9.23 kWh and from 10.44 to 17.17, respectively. According to response surface methodology, optimum condition of drying process occurred at microwave power of 630, air temperature of 70 ˚C, and air velocity of 1.2 m/s. In this optimum point, the values of diffusivity, shrinkage, specific energy consumption and total color change were 4.865×10–9, 14.22 %, 2.164 kWh and 12.312, respectively.

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In this work with a different technique we obtain upper bounds of the functional a2a4 − a 2 3 for functions belonging to a comprehensive subclass of analytic bi-univalent functions, which is defined by subordinations in the open unit disk. Moreover, our results extend and improve some of the previously known ones.

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