Tendon-driven under-actuated robotic fingers provide advantages for dexterous manipulation through reduced actuator requirements and simplified mechanical design. However, achieving both high load capacity and adaptive compliance in a compact form remains challenging. This paper presents an under-actuated tendon-driven robotic finger (UTRF) featuring a synchronous tendon routing that mechanically couples all joints with fixed angular velocity ratios, enabling the entire finger to be actuated by a single actuator. This approach significantly reduces the number of actuators required in multi-finger hands, resulting in a lighter and more compact structure without sacrificing stiffness or compliance. The kinematic and static models of the finger are derived, incorporating tendon elasticity to predict structural stiffness. A single-finger prototype was fabricated and tested under static loading, showing an average deflection prediction error of 1.0 mm (0.322% of total finger length) and a measured stiffness of 1.2x10^3 N/m under a 3 kg tip load. Integration into a five-finger robotic hand (UTRF-RoboHand) demonstrates effective object manipulation across diverse scenarios, confirming that the proposed routing achieves predictable stiffness and reliable grasping performance with a minimal actuator count.